Ramification points of homotopies: Enumeration and general theory

Andrew J. Sommese; Jonathan D. Hauenstein; Caroline Hills; Charles W. Wampler

doi:10.1515/advgeom-2025-0025

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Ramification points of homotopies: Enumeration and general theory

Andrew J. Sommese , Jonathan D. Hauenstein , Caroline Hills and Charles W. Wampler

Published/Copyright: October 28, 2025

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From the journal Advances in Geometry Volume 25 Issue 4

Abstract

Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results.

Keywords: Ramification point; branch point; polynomial system; Lefschetz pencil; homotopy continuation; numerical algebraic geometry

MSC 2010: 13P15; 65H10; 65H20

1 Introduction

The basic approach of homotopy continuation is to deform from the known solutions of a system of equations g = 0, called the start system, to compute the solutions to a related system of equations f = 0, called the target system. To make this more concrete, consider the affine setting where g = {g₁, . . . , g_N} and f = {f₁, . . . , f_N} with g_j and f_j general polynomials on ℂ^N of degree d_j ∈ ℤ_>0. Thus, both g = 0 and f = 0 have d₁ ⋅⋅ ⋅ d_N nonsingular isolated solutions, i.e., the Bézout count is sharp. A linear homotopy H : ℂ^N × ℂ→ℂ^N between g and f is

(1) H ( z , t ) = tg ( z ) + ( 1 − t ) f ( z ) .

By genericity, there are no t^∗ ∈ [0, 1] such that H(z, t^∗) = 0 has a singular solution and thus the homotopy H is said to be trackable. That is, H = 0 defines d₁ ⋅⋅ ⋅ d_N smooth solution paths z(t) : [0, 1] → ℂ^N so that H(z(t), t) ≡ 0 which smoothly connect the solutions of g = 0 at t = 1 to the solutions of f = 0 at t = 0. However, if one instead considers t ∈ ℂ rather than t ∈ [0, 1], then there is always at least one t^∗ ∈ ℂ such that H(z, t^∗) = 0 has a singular solution whenever d₁ ⋅⋅ ⋅ d_N > 1. With this formulation, it is natural to ask the following questions:

For how many t^∗ ∈ ℂ does H (z, t^∗) = 0 have a singular solution?
What kinds of singularities actually occur as singular solutions to H(z, t^∗) = 0?
For such t^∗ ∈ ℂ, what is the set of singular solutions for H(z, t^∗) = 0?

In [12], a numerical study was made of the distribution of the branch points of a general pencil of solutions of a system of quadrics in a few different projective spaces and systems of biquadrics in a few different products of projective spaces. In these special cases, the explicit systems for the singularities may be used to compute an upper bound for the number of singular solutions of a general pencil. The numerical results of that article made us realize the importance of the above questions, whose answers we did not know except for the particular systems we solved in that article. The main results of this article show that these three questions can be answered in great generality. For example, for H in (1), the answers to the above questions are:

The number of t^∗ for which H(z, t^∗) = 0 has at least one singular solution is (see (15))

d 1 ⋯ d N d 1 + ⋯ + d N − N − 1 1 d 1 + ⋯ + 1 d N + N .
For such t^∗ ∈ ℂ, each singular solution of H(z, t^∗) has multiplicity two (see Remark 24).
For such t^∗ ∈ ℂ, there is exactly one singular solution of H(z, t^∗) = 0 (see Corollary 27).

The last two are reminiscent of the classical situation for Lefschetz pencils, see e.g. [20, Chapter 2.1.1], and indeed the answers to these two questions follow from the existence of Lefschetz pencils and some standard vanishing theorems in [17].

Example 1

(Cubic pencil on ℂ). Let f(z) and g(z) be generic cubics in one variable. The results above imply that for H(z, t) = tg(z) + (1 − t)f(z), there are 4 values of t^∗ where H(z, t^∗) = 0 has a singular solution, and each singularity is a double root. Figure 1 plots (ℜ(t), ℑ(t), ℜ(z)), the real and imaginary parts of t and the real part of z, respectively, for the three sheets of H⁻¹(0). The color is based on the distance to the next nearest sheet and there is a contour plot of the minimum distance between sheets. (Distance is computed in ℂ, so zero distance implies equality of the real parts, but the converse is not necessarily true.) One sees that the sheets meet in pairs at four points. The double roots are ramification points, and the values of t^∗ where they occur are branch points.

Figure 1

A visualization of the solution set of H(z, t) = tg(z) + (1 − t)f(z) = 0 for cubics f(z) and g(z) as t varies. This solution set is a three-sheeted cover of ℂ with 4 ramification points.

The curve C ⊂ ℂ^N⁺¹ consisting of all solutions to H = 0 in (1) is smooth and connected. Smoothness is an easy consequence of Bertini’s Theorem, but connectedness requires vanishing theorems. The genus g of the smooth projective curve C‾ that C is Zariski open in satisfies

2 g − 2 = d 1 ⋯ d N d 1 + ⋯ + d N − N − 1 1 d 1 + ⋯ + 1 d N + N − 2 .

Consider the projection map π : ℂ^N⁺¹ → ℂ with π(z, t) = t. The restriction of π to C is a branched covering with d₁ ⋅⋅ ⋅ d_N sheets. The branch points are those t^∗ ∈ ℂ such that H(z, t^∗) = 0 has a singular solution, say (z^∗, t^∗), and (z^∗, t^∗) is a ramification point of order two.

In fact, all of the aforementioned results are all true even if f is not necessarily general but still has exactly d₁ ⋅⋅ ⋅ d_N nonsingular solutions. That is, the genericity of the start system g and the sharp Bézout count for f are enough. Moreover, for an arbitrary system f with the same degrees as g, many of the qualitative results still hold true. For example, if Z₀ = {(z, 0) | f(z) = 0}, then C \ Z₀ is still smooth with a unique ramification point, which has order two, over each branch point t^∗ ≠ 0.

Although H in (1) was formulated in affine space, the generality assumption ensures that the same statements hold over projective space, i.e., there is no additional structure at infinity. This paper shows that all of the above results hold in much greater generality, namely for fairly general systems on slightly singular algebraic subsets of products of projective spaces. This extra generality, which takes only slightly more care, provides the following:

there is a reduction of the generality of the linear homotopy needed for the main results to hold; and
the main results also hold for the class of normal toric varieties [10] and, in particular, for weighted projective spaces [5].

The rest of the paper is organized as follows. Section 2 considers general theory using the natural representation of a polynomial system in algebraic geometry as an algebraic section of an algebraic vector bundle. Section 3 addresses the three questions above in greater generality. Some examples are provided in Section 4. Finally, Section 5 concludes with a question for further study.

Specific examples used in this article utilized Bertini v1.6, see [3; 4], running on Intel Xeon CPU E5-2680 v3 (2.50 GHz) with 260 GB of memory. The files used for these runs plus two Maple worksheets (created using Maple 2020) are available at https://doi.org/10.7274/c.7376080.

2 Some background in algebraic geometry

The following collects some general theory for families of polynomial systems and their solutions.

A variety is defined as an irreducible and reduced algebraic set, and a projective variety is an irreducible and reduced projective algebraic set. Of the several different definitions of variety, we are following the usage in [11] and [9, Chapter 1.1]. An algebraic set M is said to be smooth if it is smooth in the scheme-theoretic sense, i.e., M is a manifold (and in particular, reduced). Given an algebraic set X, we denote its singular set by Sing (X) and its set of smooth points by X_reg = X \Sing (X). Over the complex numbers, the condition that a reduced algebraic set X is irreducible is equivalent to X_reg being connected and the closure of X_reg in the usual topology (or in the Zariski topology) being equal to X. Hence, an algebraic manifold is irreducible if and only if it is connected.

For convenience, we follow the usual convention that dim 0 = −1 and that an empty subset of a variety X has codimension dim X + 1 in X.

An algebraic vector bundle on a variety is a family of complex vector spaces whose rank equals the dimension of the vector spaces in the family. To be precise, a rank r algebraic vector bundle ℰ is defined on a variety X by using a cocycle ρ_αβ. Let GL(r, ℂ) denote the complex general linear group. Assume that there is a cover {U_α | α ∈ A} of X by Zariski open sets and algebraic maps (called a cocycle)

ρ α β : U α ∩ U β → G L ( r , C ) ∣ ( α , β ) ∈ A × A

such that

ρ_αα = 1;
if U_α ∩ U_β is nonempty, then ραβ=ρβα−1; and
if (α, β, γ) ∈ A × A × A and U_α ∩ U_β ∩ U_γ ≠ 0, then ρ_αβ ⋅ ρ_βγ = ρ_αγ.

For each U_α, let ξ_α be linear coordinates on ℂ^r (0 at the origin), whichwewrite as a vertical vector. If U_α∩U_β ≠ 0, we can patch U_α ×ℂ^r and U_β ×ℂ^r by the equation ξ_β = ρ_αβξ_α on U_α ∩ U_β. In the end, this yields a vector bundle ℰ of rank r. When r = 1, ℰ is traditionally called a line bundle. We generally use the script letters ℰ, ℱ, 𝒢 to denote vector bundles. An exception, discussed in Section 2.1, is made for the line bundles on projective space ℙ^N, where we use the notation [k]PN for integers k. An excellent place for all the details on complex vector bundles from the point of view of general mathematics, but for use in algebraic geometry is [14].

Given an algebraic vector bundle ℰ on a variety X, the dual bundle ℰ^∗ is the vector bundle with fibers equal to the dual vector space of the vector bundle fibers of ℰ → X. To define the dual bundle ℰ^∗, we need to give its cocycle. Again, we consider sets U_α × ℂ^r. We denote the linear coordinates on ℂ^r (0 at the origin) by ξˆa. As before, we regard ξˆa as a vertical vector and use ξˆa′ to denote its horizontal transpose. We use the maps

ρ ˆ α β = ρ α β − 1 ′

where Tʹ denotes the transpose of a matrix T ∈ GL(r, ℂ). Note that all the cocycle conditions hold. Moreover, we have a pairing between the fibers of ℰ^∗ and ℰ. Indeed we can send ξˆα,ξα to ξˆα′⋅ξα. Note that this is well defined, i.e., if U_α ∩ U_β ≠ 0, then on U_αβ we have

ξ ˆ β ′ ⋅ ξ β = ρ ˆ α β ⋅ ξ ˆ α ′ ⋅ ρ α β ⋅ ξ α = ξ ˆ α ′ ⋅ ρ α β − 1 ⋅ ρ α β ⋅ ξ α = ξ ˆ α ′ ⋅ ξ α .

Just as with finite-dimensional vector spaces, the dual of the dual vector bundle is the original vector bundle, i.e., (ℰ^∗) ^∗ = ℰ. A particular dual bundle of interest is the cotangent bundle TX∗ of an algebraic manifold X which is the dual vector bundle of the tangent bundle T_X.

Note that if ρ_αβ is a cocycle as above that gives rise to the vector bundle ℰ, then det(ρ_αβ) is also a cocycle which gives rise to a line bundle det(ℰ) called the determinant bundle of ℰ. This bundle may also be viewed as the rank one vector bundle whose fibers are the r^th exterior products of the fibers of ℰ. For an algebraic manifold X, the bundle KX=detTX∗ is called the canonical bundle of X.

Finally, note that if φ : X → Y is an algebraic map from one variety X to another variety Y and ℰ is an algebraic vector bundle on Y, then ℰ may be pulled back to an algebraic vector bundle, φ^∗ℰ, on X. In particular, if ρ_αβ is the cocycle (with associated open cover {U_α | α ∈ A}) giving rise to ℰ, then ρ_αβ ∘ φ (with associated open cover {φ⁻¹(U_α) | α ∈ A}) is the cocycle of φ^∗ℰ.

2.1 Sections and bundles

In this article, a polynomial system is viewed an algebraic section of an algebraic vector bundle over a projective variety X. This approach to polynomial systems matches up well with the usual notion of a polynomial system and is the approach used in [16]. For example, suppose that f = {f₁, . . . , f_k} is a system of k homogeneous polynomials in N + 1 variables, i.e., defined on ℙ^N, of degrees d₁, . . . , d_k, respectively. On ℙ^N, f_j is not a function but is an algebraic section of the algebraic line bundle djPN. Note that [0]PN is the trivial bundle, and that for all integers a, b we have

[ a ] P N ⊗ C [ b ] P N = [ a + b ] P N .

When the context is clear, the subscript ℂ in ⊗ is usually left off. Also, note that for any integer m we have

[ − m ] P N = [ m ] P N ∗

where [m]PN∗ is the dual of [m]PN. The system f is an algebraic section of

E := d 1 P N ⊕ ⋯ ⊕ d k P N ,

which is a rank k vector bundle. Systems of multihomogeneous polynomials are handled similarly.

On projective space, the algebraic line bundles [d]PN that are ample are those associated to homogeneous polynomials of positive degrees d.

Since all the bundles, manifolds, and sections we consider in this article are algebraic, outside of theorems, corollaries, and lemmas, we will usually drop the word algebraic in referring to them. Moreover, all algebraic functions on an algebraic set are functions in the usual sense of mathematics, i.e., they do not have poles and they are single-valued.

Remark 2

It should be noted that any algebraic vector bundle on ℂ^N is algebraically equivalent to the trivial bundle on ℂ^N and that algebraic functions globally defined on ℂ^N are polynomials. Therefore, an algebraic section of an algebraic vector bundle over ℙ^N reduces to a polynomial system when restricted to ℂ^N. Though most systems that arise as start systems in homotopy continuation (including those arising from polyhedral methods) come from direct sums of line bundles, there are many systems that do not correspond to such direct sums yet nevertheless fit into our approach. We work out the details of one significant example in Section 4.1.

2.2 Projectivization of a vector bundle

Let ℰ denote a rank k vector bundle on a projective variety X, i.e., an irreducible and reduced projective algebraic set. Let ℙ(ℰ) denote the space of lines in each fiber of ℰ^∗ through the origin of the fiber. This is a ℙ^k⁻¹- bundle over X with natural bundle projection π_ℙ(ℰ) : ℙ(ℰ) → X. This projective bundle over X may also be regarded as the quotient (ℰ^∗ \ X) /ℂ^x where the nonzero complex numbers ℂ^x act by the natural multiplication on each fiber of ℰ^∗.

There is a natural line bundle ξε∗ on ℙ(ℰ) which, at each point w ∈ ℙ(ℰ), is the line in πP(E)∗E∗ that w corresponds to in ℰ^∗. The dual of ξE∗, i.e. ξ_ℰ, is called the tautological line bundle on ℙ(ℰ).

An algebraic line bundle ℒ on a projective variety is called ample if there is an embedding of X in some projective space φ : X → ℙ with ℒ = φ^∗([k]_ℙ) for some positive integer k. An algebraic vector bundle ℰ is said to be ample if ξ_ℰ on ℙ(ℰ) is ample.

Given a vector bundle ℰ, let 𝒪_X(ℰ) denote the sheaf of germs of algebraic sections of ℰ.

Remark 3

Denoting this space of lines on ℰ^∗ by ℙ(ℰ) and not by (the at first sight more natural symbol) ℙ(ℰ^∗) is the standard convention in algebraic geometry; see e.g. [11, page 162]. The original convention was to use ℙ(ℰ^∗), but this fell out of favor as it often led to notational inconveniences. Nonetheless, one must remain aware of this since the older convention exists in the literature, e.g. in [9], which is cited several times herein.

Let ℰ be a vector bundle on X and let V ⊂ H⁰(X, ℰ) denote a vector subspace of the vector space H⁰(X, ℰ) consisting of all sections of ℰ on X. We say that V spans ℰ if, for each x ∈ X, the evaluation of the sections of V at x gives all points in the fiber ℰ_x of ℰ over x. When V spans ℰ, the evaluation map gives a surjective bundle map of X × V → ℰ. To make this explicit, let (x, s) ∈ X × V, then the evaluation map sends (x, s) → s(x) ∈ ℰ_x where ℰ_x is the vector space fiber over x ∈ X. The vector bundle ℰ being spanned by the space of sections V precisely means that this map is onto for each x. If ℱ^∗ is the vector subbundle of X × V whose fiber over x is

{ ( x , s ) ∈ X × V ∣ s ( x ) = 0 } ,

we have the exact sequence

(2) 0 → F ∗ → X × V → E → 0 .

Taking the duals of each vector bundle, we have the exact sequence

(3) 0 → E ∗ → X × V ∗ → F → 0.

To understand this dual sequence, consider x,e′∈Ex∗. Here eʹ is a linear map ℰ_x → ℂ taking 0 to 0. Composing with the evaluation map {x} × V → ℰ_x gives a linear map {x} × V → ℂ. This map is the element of {x} × V^∗ that (x, eʹ) goes to in (3). Similarly, given (x, sʹ) ∈ {x} × V^∗, we have the linear map sʹ : V → ℂ. Restricting to Fx∗ in (2) gives a linear map from Fx∗→C. Since the dual of the dual vector space Fx∗ is the vector space ℱ_x, we have produced the image of sʹ in ℱ_x.

If we have a line bundle ℒ on a projective variety X spanned by a vector space of sections V, then dualizing the evaluation map gives the embedding

0 → L ∗ → X × V ∗ .

This yields a map φ_V : X → ℙ(V) by sending x ∈ X to the line in V^∗ obtained from LX∗⊂{X}×V∗ by using the projection X × V^∗ → V^∗. Note that, by construction, L=φV∗[1]P(V).

The Chern classes of ℰ are cohomology classes c_j(ℰ) ∈ H^2j(X, ℤ) for j = 1, . . . , dim X. Some references for Chern classes are [9, § 3.1–3.2], [14, Chapter 1 § IV] and [19, § 11.2]. The Chern classes in [9] are more refined and defined to act on algebraic sets. Given a rank k vector bundle ℰ, the total Chern class is

c t ( E ) = 1 + c 1 ( E ) t + c 2 ( E ) t 2 + ⋯

where c^ℓ(ℰ) = 0 for ℓ > dim X.

Lemma 4

Let ℰ be a rank k algebraic vector bundle on a projective variety X spanned by a vector space V of sections. Let ℱ be as in (3) with rank e + 1 = dim V − k. Then, given an h-dimensional projective variety Y ⊂ X, we have

c 1 ξ F e + h ∩ π P ( F ) − 1 ( Y ) = c h ( E ) ∩ Y ,

where ∩ denotes the cap-product pairing of cohomology and homology.

Proof. Let s_h(ℰ) be the h^th Segre class of ℱ^∗, see e.g. [9, Chapter 3.1]. By definition,

(4) c 1 ξ F e + h ∩ π P ( F ) − 1 ( Y ) = s h F ∗ ∩ Y .

Denoting the total Segre class in [9, Chapter 3.2] by

s t F ∗ = 1 + s 1 F ∗ + s 2 F ∗ t 2 + ⋯ ,

we have c_t(ℱ^∗) = s_t(ℱ^∗)⁻¹. It follows from (2) that

s t F ∗ − 1 = c t F ∗ = c t ( E ) − 1 yielding s t F ∗ = c t ( E ) .

The result now follows from (4). □

Remark 5

Let s denote a generic section of a spanned rank N vector bundle on a projective variety X. The number of smooth isolated zeros of s is c_N(ℰ) evaluated on X. This number is familiar to people computing (multihomogeneous) Bézout numbers of start systems made up of (multihomogeneous) polynomials. We will for this reason often use the term Bézout number to refer to the number of smooth isolated zeros of a general section of a rank N spanned vector bundle on an N-dimensional projective variety.

The other Chern classes also give useful structural information about start systems of homotopies. We will see both c₁(ℰ) and c_N₋₁(ℰ) used in (12) as part of Theorem 18. The number computed by this equation gives a useful measure of the quality of a homotopy.

Recall that TX∗ denotes the cotangent bundle and KX=detTX∗ is the canonical bundle of X. Although the following is standard, we do not know a reference where the proof is provided, so we include one here for completeness.

Lemma 6

Given a rank k algebraic vector bundle ℰ on a connected algebraic manifold X, we have

K P ( ε ) = π P ( ε ) ∗ K X ⊗ det ( E ) ⊗ ξ ε − k .

Proof. Since the Jacobian of the map π_ℙ(ℰ) : ℙ(ℰ) → X is surjective, we have a short exact sequence

0 → T P ( E ) / X → T P ( E ) → π P ( E ) ∗ T X → 0 ,

where T_ℙ(ℰ)/X is the bundle of tangents to the fibers of π_ℙ(ℰ). Taking determinants, we have

(5) K P ( E ) ∗ = π P ( E ) ∗ K X ∗ ⊗ det T P ( E ) / X .

The relative version of the Euler sequence in [9, App. B5.8] (which we have adjusted for the different convention for ℙ(ℰ) in [9]) is

(6) 0 → C × P ( E ) → π P ( E ) ∗ E ∗ ⊗ ξ E → T P ( E ) / X → 0

which yields

det T P ( E ) / X = det π P ( E ) ∗ E ∗ ⊗ ξ E = det π P ( E ) ∗ E ∗ ⊗ ξ E k = π P ( E ) ∗ det ( E ) ∗ ⊗ ξ E k .

The result now follows from (5). □

Remark 7

If X is a point and V = ℰ = ℂ^N⁺¹, then P(E)=PN,ξE=[1]PN and (6) becomes

(7) 0 → [ 0 ] P N → ⨁ j = 1 N + 1 [ 1 ] P N → T P N → 0 .

From this sequence, we conclude that KPN=[−(N+1)]PN and that the total Chern class of TPN is

1 + t H 1 N + 1

where H₁ is the first Chern class of [1]PN. In particular, cjTPN=N+1jH1j.

2.3 The theorem of Bertini

There are many results in algebraic geometry loosely connected under the name Bertini’s Theorem. Some references for the results used here are [11, § III.10], [6, § 1.7] and [18, § A.9]. Among the simplest versions of Bertini’s Theorem is the algebraic version of Sard’s Theorem.

Theorem 8

(Algebraic Sard Theorem). Let f : X → Y be an algebraic map from a connected algebraic manifold to an algebraic variety Y. Assume that there is at least one point y ∈ Y such that the irreducible component Z of the fiber f ⁻¹(y) satisfies dim Z = dim X − dim Y. Then, there is a nonempty Zariski open set U ⊂ Y such that f : f ⁻¹(U) → U is onto and of maximal rank. In particular, all fibers of f over points of U are smooth with all connected components of each fiber having dimension dim Z = dim X − dim Y.

Proof. This is just [11, Corollary III.10.7] combined with the upper semi-continuity of the dimension. □

Corollary 9

Given an algebraic vector bundle ℰ on an algebraic manifold X with ℰ spanned by a finite-dimensional space of algebraic sections V, there is a nonempty Zariski open set U ⊂ V such that, for s ∈ U, Z = s⁻¹(0) is smooth and either Z is empty or codim_X Z = rank (ℰ).

There are many generalizations dealing with the situations when there are singularities. Recalling that Sing (X) denotes the singular set of an algebraic set X, we will use the following result.

Theorem 10

Let X denote a reduced algebraic set with all components of X of the same dimension, say N, and let ξ denote an algebraic line bundle on X spanned by a finite dimensional vector space of algebraic sections V. Then, there exists a nonempty Zariski open set U ⊂ V such that, for s ∈ U, D = s⁻¹(0) is either empty or

D is reduced and with all components of D having dimension N − 1;
Sing (D) ⊂ Sing (X) and D ∩ X_reg is smooth; and
each irreducible component Z of Sing (D) is a proper algebraic subset of any irreducible component of Sing (X) that Z belongs to. Hence, dim Sing(D) < dimSing (X) if Sing (D) ≠ 0.

Proof. This follows from the stronger result stated in [6, Theorem 1.7.1]. □

In Theorem 10, since the restriction of V to D spans the restriction ξ_D of ξ to D, we can repeat the above result to obtain a sequence of algebraic sets

(8) D 1 ⊂ D 2 ⊂ ⋯ ⊂ D N − 1 ⊂ D N = X

such that, if D_j ≠ 0, then the following hold:

all components of D_j have dimension j;
Sing (D_j)⊂ Sing (D_j₊₁); and
dim Sing (D_j) < dimSing (D_j₊₁) if Sing (D_j) ≠ 0.

Thus, if Sing (D_j) ≠ 0 then

dim Sing D j ≤ dim Sing ( X ) − ( N − j ) .

This gives an important fact that we state explicitly.

Corollary 11

If dim Sing (X) ≤ k, then D_j is smooth for j < N − k. In particular, if Sing (X) has codimension at least two, then D₁ is smooth.

Homotopies are typically constructed with enough conditions for vanishing theorems to guarantee that the D_j are irreducible. The following provides sufficient conditions to ensure irreducibility.

Theorem 12

Let X be an N-dimensional projective variety and let ξ be an algebraic line bundle on X spanned by a vector space V of algebraic sections. If c₁(ξ)^N ≠ 0 then, for a general s ∈ V, each D_j is irreducible for j ≥ 1.

Proof. First, assume that X is smooth and let D = s⁻¹(0) be a smooth zero set for a general s ∈ V. If D = 0, then ξ would be the trivial bundle and c₁(ξ) = 0. Hence we have D ≠ 0 with the exact sequence

(9) 0 → O X ( − D ) → O X → O D → 0.

By the Kawamata–Viehweg vanishing theorem, see [17, Corollary 7.50], we know that H⁰(X, 𝒪_X(−D)) = 0 and H¹(X, 𝒪_X(−D)) = 0. Therefore, using the long exact cohomology sequence associated to (9), dim H⁰(D, 𝒪_D) = 1. Since D is smooth, this implies it is connected. For manifolds, irreducibility is equivalent to being connected.

If X is not smooth, then let π:Xˆ→X be a desingularization. Here, Xˆ is a connected projective manifold and π gives a one-to-one and onto map from Xˆ∖π−1(Sing(X))→Xreg. Since c₁(πX^∗ξ)^N ≠ 0 and π^∗V spans π^∗ξ, a general choice of s ∈ V gives rise to the set

D ‾ = { x ∈ X ∣ s ( x ) = 0 } ⊂ X

which π maps to D such that

D‾ is connected and smooth; and
π maps Dˆ∖πD−1(Sing(D)) one-to-one and onto D_reg.

Since the image of an irreducible set is irreducible, this shows that D is irreducible.

Noting that c₁(ξ_D)^N⁻¹ = c₁(ξ)^N ≠ 0, the argument may be repeated until we reach D₁. □

When studying the number of ramification points in a fiber in Section 3.3, we will need the following when k = N − 1.

Lemma 13

Let X be an N-dimensional projective variety with dim Sing (X) ≤ k − 1. Let 𝒢 denote an algebraic vector bundle on X of rank k < N. Assume that 𝒢 is a direct sum of algebraic line bundles ℒ_j with each ℒ_j spanned by a vector space V_j of algebraic sections. Let H _X be an ample line bundle on X. Assume that, for each j ≤ k,

(10) ∏ i = 1 j − 1 c 1 L i c 1 L j 2 c 1 H X N − j − 1 ≠ 0.

Then a general section of 𝒢 from V₁ ⊕ ⋅⋅ ⋅⊕V_k has a smooth, nonempty, connected solution set of dimension N − k.

Proof. Following the methods used earlier, we may reduce to the situation where X is smooth. Note that if X was not smooth and π:Xˆ→X was desingularization as in Theorem 12, π^∗H_X won’t be ample on Xˆ, but taking any ample line bundle 𝒜 on Xˆ,HXˆ=A⊗π∗HX will be ample on Xˆ and the condition (10) will immediately yield the needed condition for j ≤ k,

∏ i = 1 j − 1 c 1 π ∗ L i c 1 π ∗ L j 2 c 1 H X ˆ N − j − 1 ≠ 0.

Let D_N = X. For each j = N − k, . . . , N − 1, let D_j denote the solution set of a general element of V_N_−j restricted to D_j₊₁. We have

X = D N ⊇ D N − 1 ⊇ ⋯ ⊇ D N − k .

The condition c1L12⋅HXN−2≠0 implies that the solution set D _N₋₁ is nonempty and thus is smooth of dimension N − 1 by Bertini’s Theorem.

By [17, Corollary 7.50] we have HrL1∗=0 for r ≤ 1. Thus from the exact sequence

0 → O L 1 ∗ → O D N → O D N − 1 → 0 ,

the long cohomology sequence yields H0ODN=H0ODN−1. Hence D_N₋₁ is connected.

Denote the restriction of H _X and ℒ_r to Dj by HDj and Lj,Dr, respectively. Noting that

c 1 L j , D N − j + 1 2 ⋅ H D N − j + 1 dim D N − j + 1 − 2 = ∏ i = 1 j − 1 c 1 L i c 1 L j 2 c 1 H X N − j − 1 ≠ 0 ,

we may continually repeat the above argument to prove the result by downward induction. □

Since the key to Lemma 13 is (10), consider what it means for (10) to fail. To that end, assume for simplicity that X in Lemma 13 has no singularities and that (10) fails for j = 1, i.e.,

c 1 L 1 2 ⋅ c 1 H X N − 2 = 0 .

Failing at other values of j is similar but the mass of indices obscures a conceptual understanding. Using Bertini’s Theorem, this implies that the solution sets of two general elements a₁, a₂ ∈ V₁ have empty intersection. Let A denote the span of a₁ and a₂. Thus, the two-dimensional vector subspace A of V₁ spans ℒ₁ under the evaluation map giving

X × A → L 1 → 0.

The map φ associated to this surjection maps X to ℙ¹ with L1=φ∗[1]P1. The solution sets of elements of A are fibers of the map. Depending on the situation, the fibers may be connected or disconnected. For example, if X = ℙ¹ × ℙ¹ and ℒ₁ was the line bundle corresponding to a bihomogeneous polynomial of bidegree (1, 0), the condition would fail and the map φ would be a product projection. Here the fibers and hence the solution sets of elements of V₁ are connected. However, if ℒ₁ was the line bundle associated to bihomogeneous polynomials of bidegree (2, 0), then φ would be a product projection composed with a degree two map of ℙ¹ to ℙ¹. In this case, the solution sets of general elements of V₁ would be disconnected (with two components). The condition given by (10) does not see which powers of the ℒ_j are used and the condition is nonzero if the connectedness is true for all the powers of the ℒ_j.

2.4 Spaces of systems and spaces of solutions

Fix a projective variety X and an algebraic vector bundle ℰ on X spanned by a vector space of algebraic sections V. Let ℙ(V) be the projective space of lines through the origin in V^∗. From (2), we have an embedding

P ( F ) → X × P V ∗ .

Let p and q denote the projections from X ×ℙ(V^∗) to X and ℙ(V^∗) respectively. Let p_ℙ(ℱ) and q_ℙ(ℱ) denote their restrictions to ℙ(ℱ). Note that p_ℙ(ℱ) is simply π_ℙ(ℱ). Each v ∈ V is a section of ℰ which, for us, is a polynomial system. Thus ℙ(V^∗) is the space of sections associated to V. For v ∈ V, let Z(v) ⊂ X denote the solution set of v. This yields the following.

Theorem 14

Given v ∈ ℙ(V^∗), one has

π P ( F ) q P ( F ) − 1 ( v ) = Z ( v ) .

In particular, this identifies the projective variety ℙ(ℱ) with the total space of all solutions corresponding to the nonzero elements of the vector space of sections V.

Proof. Fix a point v ∈ ℙ(V^∗). Choose a v̂ in V over v ∈ ℙ(V^∗). Note that the evaluation map takes {x} × v̂ to v̂(x), which is 0 if and only if x ∈ Z(v). Thus, over each of the points x ∈ X, {x} × v̂ comes from a point w_x in the fiber Fx∗ of F∗ over x if and only if v̂(x) = 0. □

Note that V^∗ spanning ℱ implies that V^∗ yields a space of sections, namely πP(F)∗V∗, which span πP(F)∗F. By the surjection πP(F)∗F→ξF,πP(F)∗V∗ spans ξ_ℱ. This is summarized in the following.

Lemma 15

The map q_ℙ(ℱ) : ℙ(ℱ) → ℙ(V^∗) is the map associated to πP(F)∗V∗ spanning ξ_ℱ and therefore we have ξF=qP(F)∗[1]PV∗.

For a map that is generically finite-to-one, the Galois/monodromy group encodes subtle structure regarding the fibers, see e.g. [13]. A transitive group means that there is a smooth path between any two points in a general fiber which can be tracked using continuation, a so-called monodromy loop. With the identifications above, we have the strong conclusion involving the transitivity of the Galois/monodromy group.

Theorem 16

Let X be a N-dimensional projective variety with dim Sing (X) ≤ N − 2. Let ℰ be a rank N algebraic vector bundle spanned by a vector space V of algebraic sections. Assume that at least one section s ∈ V has at least one isolated solution on X. Then, the map

q P ( F ) : P ( F ) → P V ∗

is generically finite-to-one and the corresponding Galois/monodromy group is transitive.

In particular, given a general line ℓ⊂PV∗,C=qP(F)−1(ℓ) is a smooth, connected algebraic curve. In fact, there is a dense Zariski open set U⊂PV∗withqP(F)−1(u) being finite and smooth for u ∈ U. Moreover, given a solution û over any u ∈ U ∩ ℓ having smooth isolated zeros, continuation using loops in U ∩ ℓ starting and ending at u will give all solutions over u.

Proof. By Theorem 8, there is a nonempty Zariski open set U ⊂ ℙ(V^∗) such that dimqP(F)−1(u)=0 for u ∈ U. Hence

c 1 ξ F dim P ( F ) = c 1 q P ( F ) ∗ [ 1 ] P V ∗ dim P V ∗ = q P ( F ) ∗ c 1 [ 1 ] P V ∗ dim P V ∗ ≠ 0 .

Corollary 11 yields smoothness of C and Theorem 12 yields connectedness of C. Transitivitiy is an immediate consequence of connectedness. □

3 Singular points of the systems parameterized by a general pencil

In this section, X is an N-dimensional projective variety with a singular set of dimension at most N − 2, and ℰ is a rank N vector bundle on a projective variety X spanned by a vector space of sections, such that at least one of the sections from V has isolated solutions. Keep in mind that statements about sections and families of sections give rise to statements about systems of polynomials and families of system of polynomials.

We shall refer to the set of sections parameterized by a line in ℙ(V^∗) as a pencil of sections. Moreover, those sections parameterized by a general line will be called a general pencil of sections.

There are three questions to ask about the singular points of the sets that the sections parameterized by a general pencil of sections vanish on.

How many singular points are there for the sections parameterized by a general pencil ℓ ⊂ ℙ(V^∗) of sections?
What are their multiplicities?
How many can there be in a fiber?

The answers to these questions constitute the main results of this paper. The first is primarily topological and answered in Section 3.2 under very general hypotheses covering practically all cases of interest for systems of polynomials arising by restriction from products of projective spaces. The last question is considered in Section 3.3.

3.1 How many singular points are there?

Let X, ℰ and V be as in Theorem 16, and let q : ℙ(ℱ) → ℙ(V^∗) be the generically finite-to-one map in Theorem 16. Let 𝒜 ⊂ ℙ(V^∗) denote the union of q (Sing (ℙ(ℱ))) and the algebraic set 𝒟 of all points y ∈ ℙ(V^∗) with dim q⁻¹(y) > 0. Let ℬ denote q⁻¹(𝒜), let 𝒱 = ℙ(V^∗) \ 𝒜 and 𝒰 = ℙ(ℱ) \ ℬ. This gives rise to the following.

Lemma 17

With this setup, 𝒰 is smooth and the map q_𝒰 : 𝒰 → 𝒱 is finite-to-one. Then dim 𝒜 ≤ dim ℙ(V^∗)− 2 and thus, a general line ℓ ⊂ ℙ(V^∗) lies in 𝒱. Moreover, given an arbitrary point y ∈ ℙ(V^∗) and a general point x ∈ ℙ(V^∗), the line containing x and y meets 𝒜 in at most y.

Proof. The statements about smoothness and q_𝒰 being finite-to-one are true by the construction of 𝒱 and 𝒰. To see that dim 𝒜 ≤ dim ℙ(V^∗)− 2, it suffices to show that dim q (Sing (ℙ(ℱ))) ≤ dim ℙ(V^∗)− 2 and dim 𝒟 ≤ dim ℙ(V^∗)− 2.

Since dim Sing (X) ≤ dim X − 2, we know dim Sing (ℙ(ℱ)) ≤ dim ℙ(ℱ) − 2. Additionally, dim ℙ(ℱ) = dim ℙ(V^∗) yields

dim q ( Sing ( P ( F ) ) ) ≤ dim Sing ( P ( F ) ) ≤ dim P ( F ) − 2 = dim P V ∗ − 2 .

It follows from dim 𝒟 < dim q⁻¹(𝒟) < dim ℙ(ℱ) = dim ℙ(V^∗) that dim 𝒟 ≤ dim ℙ(V^∗)− 2.

The fact that a general line containing y meets 𝒜 in at most y follows by dimension counting. Consider the closure Q of the union of all lines on ℙ(V^∗) containing y meeting𝒜 in at least one point distinct from y. We have dim Q ≤ dim 𝒜 + 1 ≤ dim X − 1. □

Let ℓ be a general line in ℙ(V^∗) which lies in 𝒱. By Bertini’s Theorem, C = q⁻¹(ℓ) is smooth, and by Theorem 16, C is connected. The line bundle associated to the ramification locus R of q_𝒰 is identified by [9, Ex. 3.2.20] with KU⊗q∗KV∗. Note that R ∩ C are the ramification points of q_C. A straightforward check shows that, for x ∈ C ∩ R, the multiplicity of the component of R at x is the ramification index of q_C at x. In particular, over 𝒰, R is the zero set of a section of the line bundle

R := K P ( F ) ⊗ q P ( F ) ∗ K P V ∗ ∗ .

It immediately follows that deg c₁(ℛ)_C = C ∩ R. Note that the singular points of a system y ∈ ℓ are the points in qC−1(y)∩R. Additionally, since KPV∗=[−dimV]PV∗ and ξF=qP(F)∗[1]P(V)∗, we conclude from Lemma 6 that

(11) R = π P ( F ) ∗ K X ⊗ det ( F ) ⊗ ξ F N .

The following counts singular points with respect to the ramification index. Thus, for example, a multiplicity two singular point of a system contributes one to the count.

Theorem 18

Let X be an N-dimensional projective variety with singular set of codimension at least two. Let ℰ be a rank N vector bundle spanned by a vector space V of sections. Assume that at least one section s ∈ V has at least one isolated solution on X. Then the number of singular points of solutions of the sections parameterized by a general line ℓ ⊂ ℙ(V^∗) is

(12) c 1 K X + c 1 ( E ) c N − 1 ( E ) + N c N ( E ) .

In particular, when ℰ is a direct sum of line bundles ℒ₁ ⊕⋅ ⋅⋅ ⊕ ℒ _N, this number equals

(13) c 1 K X + ∑ j = 1 N c 1 L j ∑ i = 1 N ∏ j ≠ i c 1 L j + N ∏ j = 1 N c 1 L j .

Proof. Letting C = q⁻¹(ℓ), the number of singular points is

R ∩ C = π P ( F ) ∗ c 1 K X + c 1 ( det ( F ) ) + N c 1 ξ F ⋅ c 1 ξ F dim P ( F ) − 1 .

Using Lemma 4, we have

R ∩ C = π P ( F ) ∗ c 1 K X + c 1 ( det ( F ) ) ⋅ c N − 1 ( E ) + N c N ( E ) .

This immediately implies (12) with (13) trivially following from (12). □

The following specializes Theorem 18 to a product of projective spaces.

Corollary 19

Let X=Pa1×⋯×Pak be a product of projective spaces and N = a₁ + ⋅⋅ ⋅ + a_k. For j = 1, . . . , N, let ℒ_j be a line bundle of multidegrees d_j_,1, . . . , d_j_,k and ℰ = ℒ₁ ⊕⋅ ⋅⋅ ⊕ ℒ_N. Let V be a vector space of sections of ℰ that spans ℰ and assume that at least one section s ∈ V has at least one isolated solution on X. For j = 1, . . . , N, let δ_j be the linear function ∑i=1kdj,iHi in the variables H_i . Then the number of singular points of solutions of the systems parameterized by a general line ℓ ⊂ ℙ(V^∗) is the coefficient of H1a1⋯Hkak in the polynomial

(14) ∑ j = 1 N δ j − ∑ i = 1 k a i + 1 H i δ 1 ⋯ δ N 1 δ 1 + ⋯ + 1 δ N + N δ 1 ⋯ δ N .

In particular, when X = ℙ^N and E=d1PN⊕⋯⊕dNPN, then this number equals

(15) d 1 + ⋯ + d N − N − 1 d 1 ⋯ d N 1 d 1 + ⋯ + 1 d N + N d 1 ⋯ d N

Proof. Let πj:X→Paj be the projection of X onto its j^th factor and let Hj=c1π∗[1]Paj. Note that for a vector (b₁, . . . , b_k) of nonnegative integers such that b₁ + ⋅⋅ ⋅ + b_k = N, the term H1b1⋯Hkbk evaluated on X equals 1 if b_j = a_j for all j = 1, . . . , k and equals zero otherwise. With this, the result immediately follows from Theorem 18. □

Remark 20

One can easily use (14) to count the number of singular points for product of projective spaces using a computer algebra system. As mentioned at the end of the Introduction, we have developed a Maple worksheet for this. The following summarizes the steps used to perform this calculation following the notation in Corollary 19 and its proof. First, the total Chern class is computed via

c t ( E ) = ∏ j = 1 N 1 + t c 1 L j .

The Bézout number is the coefficient of t^N in c_t(ℰ), which is a polynomial in the H_j ’s. In fact, since c_N(ℰ) is simply a coefficient multiplied by H1a1⋯Hkak, the Bézout number is precisely the coefficient of H1a1⋯HNak. In the Maple worksheet, this is computed in stages by first computing the coefficient, say A₁ of H1a1, and then taking the coefficient, say A₂ of H2a2 in A₁, and so on.

Next, c_N₋₁(ℰ) is equal to the coefficient of t^N⁻¹ in the polynomial c_t(ℰ). Moreover, from (14) one obtains

c 1 K X = − ∑ j = 1 N a j + 1 H j .

Thus, the Maple worksheet computes the product

c 1 K X + c 1 ( E ) c N − 1 ( E )

and extracts the coefficient of H1a1⋯HNaN in this expression.

Finally, the two coefficients are combined via (12) to yield the number of singular points.

The following defines the ratio of the number of singularities in a general pencil with the Bézout number, which is one measure of the quality of a homotopy.

Definition 21

Let B be the Bézout number and let σ be the number of singularities in a general pencil. Then the ratio r = σ/B is called the singularity ratio.

Hence for a homotopy, the singularity ratio is equal to the number of singular points per solution path that needs to be tracked. A high singularity ratio means that there are many ramification points in relation to the number of solution paths to track. Similarly, a low singularity ratio means that there are few ramification points in relation to the number of solution paths to track.

We close this subsection with two examples.

3.1.1 Cyclic example

Let X = (ℙ¹)^N be a product of N copies of ℙ¹’s with N ≥ 2. Fix multidegrees d₁, . . . , d_N with d_j = (d_j_,1, . . . , d_j_,N) such that

d j , k = 1 if k = j or k = j + 1 m o d N 0 otherwise .

Consider the general pencil λf + μg where f = (f₁, . . . , f_N) and g = (g₁, . . . , g_N) are general systems of multihomogeneous polynomials on X such that both f_j and g _j have multidegree d_j . The corresponding line bundles ℒ_j and the vector bundle ℰ are clear. Moreover, it is straightforward to check that the coefficient of H₁ ⋅⋅ ⋅ H_N in

H 1 + H 2 ⋯ H N + H 1

is 2. So the Bézout number of systems in this pencil is B = 2.

Since det ℰ is the line bundle of multidegrees (2, . . . , 2) and K _X is a line bundle of multidegrees (−2, . . . , −2), K _X ⊗det ℰ has multidegrees all 0. Hence c₁ (K_X)+ c₁ (det ℰ) = 0 and so the number of singular points of solutions of the systems parameterized by the general pencil is σ = 2N. Thus the singularity ratio is r = σ/B = N, and by Hurwitz’s formula, the genus g of the smooth curve of all the solutions of the pencil satisfies 2g − 2 = −4 + 2N, i.e., g = N − 1.

3.1.2 Polynomials with the same multidegrees

Consider a further simplification of Corollary 19 when every polynomial is assumed to have the same multidegree on X=Pa1×⋯×Pak. The cases of k = 1 and k = 2 with the same multidegree were considered in [12, Theorems 2 and 4], respectively, using multihomogeneous counts. The following considers the case when k is arbitrary and recovers the k = 1 and k = 2 counts as special cases.

To that end, let N = a₁ + ⋅⋅ ⋅ + a_k and let ℰ be a direct sum ℒ₁ ⊕ ⋅⋅ ⋅⊕ ℒ_N where all of the line bundles ℒ_j have the same multidegree (d₁, . . . , d_k). Then the number of singularities in a general pencil is

(16) N a 1 , … , a k d 1 a 1 ⋯ d k a k N ( N + 1 ) − ∑ j = 1 k a j a j + 1 d j

where Na1,…,ak is the usual multinomial coefficient, namely

N a 1 , … , a k = N ! a 1 ! ⋯ a k ! .

To see this, let H_j be as in Corollary 19, i.e., the first Chern class of the line bundle of multidegrees 0 in all places but the j^th where it is degree 1. Then we have

c 1 K X ⊗ det ( E ) = ∑ j = 1 k N d j − a j − 1 H j , c N − 1 ( E ) = N ∑ j = 1 k d j H j N − 1 , c N ( E ) = ∑ j = 1 j d j H j N .

As in the proof of Corollary 19, if (b₁, . . . , b_k) are nonnegative integers summing to N, then

H 1 b 1 ⋯ H k b k = 1 a 1 , … , a k = b 1 , … , b k , 0 otherwise .

As always, we identify the generator of H^2N(X,ℤ) = ℤ corresponding to X with 1. Then we have

c 1 K X ⊗ det ( E ) c N − 1 ( E ) = N ∑ j = 1 k N d j − a j − 1 ( N − 1 ) ! a j − 1 ! ∏ i ≠ j a i ! d j a j − 1 ∏ i ≠ j d i a i and c N ( E ) = N a 1 , … , a k d 1 a 1 ⋯ d k a k .

Thus (12) becomes

(17) N ∑ j = 1 k N d j − a j − 1 ( N − 1 ) ! a j − 1 ! ∏ i ≠ j a i ! d j a j − 1 ∏ i ≠ j d i a i + N N a 1 , … , a k d 1 a 1 ⋯ d k a k = N N a 1 , … , a k d 1 a 1 ⋯ d k a k ∑ j = 1 k N d j − a j − 1 a j N d j + 1 ,

which is equivalent to (16) since N = a₁ + ⋅⋅ ⋅ + a_k.

When k = 2, (16) reduces to the formula provided as an upper bound in [12, Theorem 4]. In particular, with N = a₁ + a₂ the multinomial coefficient becomes the binomial coefficient yielding

(18) a 1 + a 2 a 1 d 1 a 1 d 2 a 2 a 1 + a 2 a 1 + a 2 + 1 − a 1 a 1 + 1 d 1 − a 2 a 2 + 1 d 2

which is equal to the formula reported in [12, Theorem 4], namely

(19) 2 a 1 + a 2 a 1 d 1 a 1 − 1 d 2 a 2 − 1 a 1 + a 2 + 1 2 d 1 d 2 − a 1 + 1 2 d 2 − a 2 + 1 2 d 1 .

Finally, consider the case when a₁ = ⋅⋅ ⋅ = a_k = a and d₁ = ⋅⋅ ⋅ = d_k = d, i.e., polynomials of multidegree (d, . . . , d) on X = (ℙ^a)^k with N = k ⋅ a. Then (16) and (17) become

(20) N N ! ( a ! ) k d N k ( N d − a − 1 ) a N d + 1 = N ! ( a ! ) k d N N ( N + 1 ) − N ( a + 1 ) d .

When k = 1, i.e., N = a, (20) reduces to the formula provided in [12, Theorem 2], namely

N d N N d − N − 1 d + 1 = ( N + 1 ) N ( d − 1 ) d N − 1 = 2 N + 1 2 ( d − 1 ) d N − 1 .

Moreover, since d^N is the Bézout number, the singularity ratio is

r = N N d − N − 1 d + 1 = N ( N + 1 ) 1 − 1 d .

Thus the singularity ratio grows quadratically in N when d ≥ 2.

3.2 What are the multiplicities of the singular points?

The short answer is that singularities almost always have multiplicity two.

Theorem 22

Let X be a projective variety with singularities of codimension at least two. Let ℰ = 𝒢 ⊕ ℒ where 𝒢 is a rank N − 1 vector bundle on X and ℒ is a line bundle on X. Let V = V_𝒢 ⊕ V_ℒ where V_𝒢 spans 𝒢, V_ℒ spans ℒ, and the associated map of X to ℙ(V_ℒ) is an embedding. Assume that at least one section of ℰ coming from V has isolated solutions. Let ℓ be a general line on ℙ(V^∗). Then the following hold:

the curve C of the solutions of the systems parameterized by ℓ is smooth and connected; and
the singularities of the systems parameterized by ℓ (which are the ramification points of the projection C → ℓ) are of multiplicity two (and ramification index one).

Proof. The first statement was shown in Theorem 16.

Since the multiplicity two condition is an open condition, it suffices to show that this is true for one system. Choose a general element g of V_𝒢. By the fact that V_𝒢 spans 𝒢 and Bertini’s Theorem, the solution set 𝒞 of g is either smooth of dimension one or empty. It cannot be empty since a general element of V has isolated solutions. Thus 𝒞 is a union of a finite number of smooth curves 𝒞₁, . . . , 𝒞_k. By the existence of Lefschetz pencils, see e.g. [20, Chapter 2.1.1], it follows that for almost every choice of two sections ℓ₂, ℓ₃ of V_ℒ, the singularities of the systems on 𝒞 parameterized by

λ ℓ 2 + μ ℓ 3 ∣ [ λ , μ ] ∈ P 1

have multiplicity two. Therefore, the linear ℙ¹ on ℙ(V^∗) of systems

g λ ℓ 2 + μ ℓ 3

has the property that all systems parameterized by the ℙ¹ have multiplicity two solutions. □

Corollary 23

Under the hypotheses of Theorem 22, the multiplicities of the components of R are all one. Therefore, by Lemma 15, given an arbitrary point y ∈ ℙ(V^∗) and a general x ∈ ℙ(V^∗), the singularities of the systems parameterized by the line containing x and y on ℙ(V^∗) have multiplicity two with the possible exception of the singular points of the system y.

Remark 24

These assumptions in Theorem 22 are actually quite mild. For example, the hypotheses hold on ℙ^N when the degrees d₁, . . . , d_N satisfy d₁ ⋅⋅ ⋅ d_N > 0. More generally, the conditions of the theorem hold on Pa1×⋯×Pak with N = a₁ + ⋅⋅ ⋅ + a_k equations having corresponding multidegrees d₁ = (d_1,1, . . . , d_1,k), . . . , d_N = (d_N_,1, . . . , d_N_,k) satisfying

there exists i ∈ {1, . . . , N} such that d_i_,j > 0 for every j = 1, . . . , k; and
a general system with these multidegrees has isolated solutions.

3.3 What is the maximum number of singularities in a fiber?

Consider the hypotheses of Theorem 22 and the curve 𝒞 arising in its proof. If we knew that 𝒞 was connected, then it would immediately follow that there was at most one singular solution in a fiber of a general pencil of systems. For instance, if besides being spanned, suppose that X was smooth and 𝒢 was ample of rank N −1, then this would follow from a generalization of the first Lefschetz Theorem, see e.g. [15, Theorem 7.1.1]. For direct sums of line bundles, the following is much stronger.

Theorem 25

Let X be an N-dimensional projective variety with singular set of codimension at least two and let H _X be an ample line bundle on X. Let ℰ = ℒ₁ ⊕⋅ ⋅⋅ ⊕ ℒ _N where each ℒ_j is a line bundle on X with V_j spanning ℒ_j . Assume that the map of X to ℙ(V_N) associated to ℒ _N is an embedding on X_reg. Let V = V₁ ⊕⋅ ⋅⋅ ⊕ V_N and assume that, for j = 1, . . . , N − 1,

(21) ∏ i = 1 j − 1 c 1 L i c 1 L j 2 c 1 H X N − j − 1 ≠ 0 .

Then there is at most one singularity for every system parameterized by a general line on ℙ(V^∗).

Proof. A Lefschetz pencil, see [20, Chapter 2.1.1], has at most one singularity in a fiber. As noted above, if the curve 𝒞 in the proof of Theorem 22 was connected, we would have the requisite conclusion. For this, one can take 𝒢 = ℒ₁ ⊕⋅ ⋅⋅ ⊕ ℒ_N₋₁ and utilize Lemma 13. □

In terms of R, the ramification locus of the map q_𝒰 : 𝒰 → 𝒱, Theorem 25 says that two distinct components of R cannot have the same image under q. This yields the following.

Corollary 26

Under the hypotheses of Theorem 25, given an arbitrary point y ∈ ℙ(V^∗) and a general point x ∈ ℙ(V^∗), there is at most one singularity for any of the systems parameterized by the line containing x and y on ℙ(V^∗) with the possible exception of the singular points of the system y.

The following is an immediate consequence when applied to ℙ^N.

Corollary 27

Let E=d1PN⊕⋯⊕dNPN and V = H⁰(ℰ). Then the conditions of Theorem 25 and Corollary 26 hold if (and only if ) d_j > 0 for j = 1, . . . , N.

As noted in the discussion after Lemma 13, the failure of the condition imposed by (21) does not preclude the result from still being true. Thus, this is a sufficient condition.

When considering a product of projective spaces, (21) is very natural and can be easily checked using a computer algebra system. Similar with Remark 20, we have created a Maple worksheet for performing this check as well.

Although (21) holds for many common situations, see e.g. Corollary 27, it is interesting to consider cases for which it does not hold. Even when using all the sections of the line bundles, the Chern class condition (21) for connectedness can fail and/or ℒ _N can fail to be ample. The following considers two such cases.

Example 28

The following two cases are situations in which the condition in (21) is violated and, subsequently, the conclusion of Theorem 25 fails to be true. The cases are general pencils on X = ℙ¹ × ℙ¹ with

bihomogeneous polynomials of bidegrees (d, 0) and (0, d); and
bihomogeneous polynomials of bidegrees (3, 0) and (2, 2).

In the first case, Theorem 18 yields that the total number of singularities of the solution set (all double points with ramification index one) is σ_d = 4d(d − 1) with the Bézout number being d². The case d = 1 has the Bézout number one (as can easily be seen directly) and σ₁ = 0, i.e., no singularities, and thus is not interesting. Using Bertini, see [3; 4], we obtain the following in under a second:

for d = 2: σ₂ = 8 = 4 ⋅ 2 as predicted by the formula, but the singularities arise from four pairs, i.e. there are four fibers with singularities and each has two singular points; and
for d = 3, σ₃ = 24 = 8 ⋅ 3 as predicted by the formula, but the singularities arise from eight triplets, i.e. there are eight fibers with singularities and each has three singular points.

In the second case, ℒ₂ is ample and Theorem 18 yields that the total number of singularities of the solution set (all double points with ramification index one) is σ = 18 with the Bézout number of the system being 6. Using Bertini, see [3; 4], we obtain 14 fibers containing at least one singularity in under a second, with the following providing a summary of these 14 fibers:

10 of these fibers contain one singularity; and
4 of these fibers contain two singularities.

Note that σ = 18 = 10 + 4 ⋅ 2.

Remark 29

The second case in Example 28 shows that the assumption of connectedness in the existence theorem for Lefschetz pencils cannot be dropped.

4 Examples

The following collects some interesting examples. The first, in Section 4.1, considers the sections of the tangent bundle of ℙ^N twisted by [d]PN, i.e., TPN(d)=TPN⊗C[d]PN, for d≥−1. This is an example of a bundle that is not a direct sum of line bundles. It leads to systems with lower Bézout numbers than one would expect using multihomogeneous counts.

The second, in Section 4.2 considers Alt’s nine-point path synthesis problem for four-bar linkages; see [1; 21]. This problem has had a major influence on our view of continuation and ramification points. The realization in [2, § 3.3] that 0.83% of the paths of the homotopy to solve it passed near enough to singularities of the pencil to require precision higher than double was one of the inspirations of this article and [12]. Therefore, the numbers of singularities for systems like this are of particular interest to us. In particular, Table 1 compares four different formulations.

Section 4.3 considers a closely related family, namely the Alt–Burmester synthesis problems for four-bar linkages; see [7]. Table 2 compares two different formulations on the collection of Alt–Burmester problems.

4.1 Twists of the tangent bundle of projective space

The tangent bundle TPN of PN is not a direct sum of line bundles, as can be checked from the Chern classes of the bundle computed in Remark 7. Nonetheless, it gives rise to a very interesting class of polynomial systems.

Theorem 30

Let (x₁, . . . , x_N) be coordinates on ℂ^N and let d ≥ −1 be an integer. Consider systems of the form

(22) p 1 ( x ) − x 1 q ( x ) ⋮ p N ( x ) − x N q ( x )

where q(x) is a homogeneous polynomial of degree d + 1 in x₁, . . . , x_N and each p_j(x) is a polynomial (not necessarily homogeneous) of degree d + 1. Each such system extends to an algebraic section of TPN(d), the tangent bundle of ℙ^N twisted by a integer, i.e. TPN⊗C[d]PN. Moreover, the correspondence between these systems and the sections of TPN(d) is one-to-one and onto.

Proof. To see this, fix coordinates (z₀, . . . , z_N) on ℂ^N^{+ 1}. Consider the usual map

π : C N + 1 ∖ { ( 0 , … , 0 ) } → P N given by z 0 , … , z N → z 0 , z 1 , … , z N

where we regard the [z₀, z₁, . . . , z_N] as homogeneous coordinates. The vector fields on ℂ^N⁺¹ that are mapped to vector fields on ℙ^N are precisely those of the form

(23) ∑ j = 0 N L j ( z ) ∂ ∂ z j

where L_j(z) is homogeneous linear in the z variables. The only one of these vector fields that goes to zero on ℙ^N is

Z 0 ∂ ∂ Z 0 + ⋯ + Z N ∂ ∂ Z N

which is tangent to the fibers of the map π and goes to 0.

Consider the long exact cohomology sequence associated to (7) tensored with [d]PN. On ℙ^N we have HjPN,[k]PN=0 for all k if 0 < j < N. Thus we have

0 → H 0 P N , [ d ] P N → ⨁ j = 0 N H 0 P N , [ d + 1 ] P N → H 0 P N , T P N ( d ) → 0.

The first term is exactly the vector space of homogeneous polynomials of degree d. The second term is the vector space of (N + 1)-tuples of homogeneous polynomials of degree d + 1, with the map from the first to the second term given by

g ( z ) → Z 0 g ( z ) , … , z N g ( z ) .

The third term are the sections of the algebraic bundle TPN(d). The map from the second term to the third term is given by

p 0 ( z ) , … , p N ( z ) → p 0 ( z ) ∂ ∂ z 0 + ⋯ + p N ( z ) ∂ ∂ z N .

Exactness comes down to z0∂∂z0+⋯+zN∂∂zN being the only vector field of those in (23) that is zero on ℙ^N.

We map ℂ^N → ℙ^N by sending (x₁, . . . , x_N)→ [1, x₁, . . . , x_N]. We will now proceed to see what the sections p0∂∂z0+⋯+pN∂∂zN give rise to when restricted to this ℂ^N. Given

p 0 ( z ) ∂ ∂ z 0 + ⋯ + p N ( z ) ∂ ∂ z N ,

we can subtract

g ( z ) z 0 ∂ ∂ z 0 + ⋯ + z N ∂ ∂ z N

where g(z) is of degree d without changing the vector field on ℙ^N. In this way, we can assume that p₀(z) is homogeneous of degree d+ 1 in the variables z₁, . . . , z_N and the p_j(z) for j = 1, . . . , N are arbitrary homogeneous polynomials of degree d + 1. Using

z 0 p 0 ( z ) ∂ ∂ z 0 = − z 1 p 0 ( z ) ∂ ∂ z 1 ⋯ − z N p 0 ( z ) ∂ ∂ z N ,

we see that the algebraic sections of TPN(d) are precisely of the form

p 1 ( z ) − z 1 p 0 ( z ) ∂ ∂ z 1 + ⋯ + p N ( z ) − z N p 0 ( z ) ∂ ∂ z N .

On ℂ^N, with the x coordinates obtained via x_i = z_i/z₀ with z₀ = 1, these are exactly the systems described by (22). □

The Bézout number for TPN(d) is

(24) B T P N ( d ) := ∑ j = 0 N N + 1 j d N − j = ( d + 1 ) N + 1 − 1 d = ∑ j = 0 N ( d + 1 ) j

which is precisely the number of isolated smooth solutions of a generic algebraic section of TPN(d). In (24), the first equality follows from combining the formulae for the Chern classes of TPN, see e.g. Remark 7, and the formula for the Chern classes of a bundle twisted by a line bundle. The solution count of (24) is lower than the count of

( d + 2 ) N = ( ( d + 1 ) + 1 ) N = ∑ j = 0 N N j ( d + 1 ) j

for N general degree d + 2 polynomials. In [16, page 145] a different, but incorrect, system of two degree d + 2 polynomials is stated to have the properties of the above system when N = 2.

Further,

c N − 1 T P N ( d ) = ∑ j = 0 N − 1 ( N − j ) N + 1 j d N − 1 − j H 1 N − 1 = ( N d − 1 ) ( d + 1 ) N + 1 d 2 H 1 N − 1 ,

where H1=c1[1]PN. This follows from the same sort of algebra as the computation of the Bézout number above. An easier computation shows that

c 1 K P N + c 1 T P N ( d ) = N d H 1 .

Thus, Theorem 18 shows that a general pencil of systems has

(25) σ T P N ( d ) := N d ( N d − 1 ) ( d + 1 ) N + 1 d 2 + N ( d + 1 ) N + 1 − 1 d = N ( N + 1 ) ( d + 1 ) N

singularities. In particular, the singularity ratio is

σ T P N ( d ) B T P N ( d ) = N ( N + 1 ) d ( d + 1 ) N ( d + 1 ) N + 1 − 1 = N ( N + 1 ) d d + 1 + O 1 ( d + 1 ) N

where as usual O1(d+1)N is Hardy’s O, i.e., the remainder to the approximation N(N+1)dd+1 bounded by a constant independent of d (depending on N) times 1(d+1)N.

Compare this to the more general case of systems of N polynomials of degree d + 2. The bundle is

G = ⨁ j = 1 N [ d + 2 ] P N

The Bézout number is B_𝒢 = (d + 2)^N and number of singularities is σ_𝒢 = (d + 2)^N⁻¹N(N + 1)(d + 1). We already saw that B_𝒢 is larger than BTPN(d). Similarly, σ_𝒢 is always larger than σTPN(d) since

σ G σ T P N ( d ) = d + 2 d + 1 N − 1 .

In particular,

σ G D G σ T P N ( d ) D T P N ( d ) = 1 + 1 − 1 ( d + 1 ) N − 1 ( d + 2 ) d > 1 .

4.2 Alt’s nine-point path synthesis problem

To keep notation simple, we use a compact description of the systems and leave details regarding the actual systems to [21]. The first formulation of Alt’s problem consists of solving sixteen cubics and eight quartics on ℙ²⁴, which will be represented by

( 3 H ) 16 ( 4 H ) 8 on P 24 .

The Freudenstein and Roth formulation, which was the starting point in [21], yields a system of eight septics on ℙ⁸, which will be represented by

( 7 H ) 8 on P 8 .

In [21], four new variables were added to the Freudenstein and Roth formulation along with four new polynomials which reduced the system to four quadrics and eight quartics on ℙ¹². We represent this by

( 2 H ) 4 ( 4 H ) 8 on P 12 .

Finally, one can view this system naturally on ℙ⁶ × ℙ⁶ consisting of

two bihomogeneous polynomials of multidegree (2, 0);
two bihomogeneous polynomials of multidegree (0, 2); and
eight bihomogeneous polynomials of multidegree (2, 2).

This will be represented by

2 H 1 2 2 H 2 2 2 H 1 + 2 H 2 8 on P 6 × P 6 .

With these four formulations of Alt’s problem, Table 1 summarizes the Bézout numbers (B), the numbers of singularities (σ), and the singularity ratios (r = B/σ).

Table 1

Summary of different formulations of Alt’s nine-point path synthesis problem

Version	Degree Structure	B	σ	r = σ/B
Original	(3H)¹⁶(4H)⁸ on ℙ²⁴	11,019,960,576	4,275,744,703,488	388
Freudenstein–Roth	(7H)⁸ on ℙ⁸	5,764,801	355,770,576	≈ 61.7
Total Degree	(2H)⁴(4H)⁸ on ℙ¹²	1,048,576	125,829,120	120
Bihomogeneous	(2H₁)²(2H₂)²(2H₁ + 2H₂)⁸ on ℙ⁶ × ℙ⁶	286,720	31,768,576	110.8

The only nontrivial one of these is the bihomogeneous formulation. Letting [a, b] be the line bundle of bidegree (a, b) on ℙ⁶ × ℙ⁶, we have

E = [ 2 , 0 ] ⊕ 2 ⊕ [ 0 , 2 ] ⊕ 2 ⊕ [ 2 , 2 ] ⊕ 8 .

The total Chern class is

c t ( E ) = 1 + 2 H 1 t 2 1 + 2 H 2 t 2 1 + 2 H 1 + 2 H 2 t 8 .

Thus c₁(det(ℰ)) = c₁(ℰ) is equal to the coefficient of t in c_t(ℰ), namely 20(H₁ + H₂). Similarly, c₁₁(ℰ) is equal to the coefficient of t¹¹. Remembering that H1jH212−j equals 1 if j = 6 and 0 otherwise, we have

c 11 ( E ) = 1 , 089 , 536 H 1 5 H 2 5 H 1 + H 2 .

Similarly, the Bézout number is c₁₂(ℰ) which is equal to 286,720 under the usual identification for connected compact manifolds M of HdimRM(X,Z) with ℤ. The canonical bundle K _X equals [−7, −7] and therefore c₁(K_X) = −7H₁ − 7H₂. Putting everything together with Theorem 18 yields

c 1 K X + c 1 ( E ) c 11 ( E ) + 12 c 12 ( E ) = 13 H 1 + H 2 ⋅ 1 , 089 , 536 H 1 5 H 2 5 H 1 + H 2 + 12 ⋅ 286 , 720 = 14 , 163 , 968 ⋅ H 1 5 H 2 5 H 1 + H 2 2 + 3 , 440 , 640 = 14 , 163 , 968 ⋅ 2 + 3 , 440 , 640 = 31 , 768 , 576 .

For all four formulations summarized in Table 1, using all sections of ℰ in each case, the conditions of Theorem 25 and Corollary 26 are satisfied.

4.3 Alt–Burmester systems

Alt’s nine-point path synthesis problem [1] in Section 4.2 and Burmester’s five-pose path synthesis problem in [8] can be considered as part of a family of four-bar synthesis problems called Alt–Burmester problems; see [7]. This family of zero-dimensional problems is parameterized by nonnegative integer pairs (m, n) such that 2m+ n = 10 with m ≥ 1. The (m, n) synthesis problem aims to compute four-bar linkages satisfying m poses (position and orientation) and n precision points (position only). The reason for m ≥ 1 is that one can always trivially match any orientation at one point simply by setting the frame of reference. In particular, the two extremes of the Alt–Burmester problems are Burmester’s problem corresponding with (5, 0) and Alt’s problem corresponding to (1, 8), where an orientation is added to one of the nine points to trivially set the frame of reference.

While the formulation in Section 4.2 was highly specialized to Alt’s problem, here we follow the “standard” formulation from [7]. In this “standard” formulation, an (m, n) Alt–Burmester problem (with 2m + n = 10) corresponds with a system with degree structure denoted as

∏ j = 1 n H 2 j − 1 + H 2 j + H 2 n + 1 + H 2 n + 2 H 2 j − 1 + H 2 j + H 2 n + 3 + H 2 n + 4 H 2 j − 1 + H 2 j ⋅ H 2 n + 1 + H 2 n + 2 m − 1 H 2 n + 3 + H 2 n + 4 m − 1 on P 1 2 n × P 2 4 .

Note that 2n + 8 = 3n + 2(m − 1). In this formulation, rotations of the coupler link at each precision point are variables of the system, cast onto ℙ¹ × ℙ¹ for each of the n precision points via isotropic coordinates. When n ≥ 1, the corresponding rotation variables can easily be eliminated to produce an “alternate” formulation with degree structure

2 H 1 + 2 H 2 + 2 H 3 + 2 H 4 n H 1 + H 2 m − 1 H 3 + H 4 m − 1 on P 2 4 .

In particular, the (1, 8) “standard” and “alternate” systems provide two more presentations of Alt’s nine-point problem. The Bézout number for both of these formulations is 645,120 which falls between the last two entries in Table 1.

For concreteness, consider the (4, 2) problem. The “standard” formulation degree structure is

(26) H 1 + H 2 + H 5 + H 6 H 1 + H 2 + H 7 + H 8 H 1 + H 2 ⋅ H 3 + H 4 + H 5 + H 6 H 3 + H 4 + H 7 + H 8 H 3 + H 4 ⋅ H 5 + H 6 3 H 7 + H 8 3 on P 1 4 × P 2 4

with (27) providing explicit polynomials in affine coordinates for simplicity. Meanwhile, the “alternate” formulation degree structure is

2 H 1 + 2 H 2 + 2 H 3 + 2 H 4 2 H 1 + H 2 3 H 3 + H 4 3 on P 2 4 .

Table 2 provides the Bézout number (B), the number of singularities (σ), and the singularity ratio (r = σ/B) for each of the zero-dimensional Alt–Burmester systems. Note that the Bézout number is the same for both the “standard” and “alternate” formulations, but the “alternate” formulation has fewer singularities in an ambient space of smaller dimension.

Table 2

Comparison of “standard” and “alternate” formulations of the zero-dimensional Alt–Burmester problems

(m, n)	Formulation	B	σ	r = σ/B
(5, 0)	Standard	36	576	16
(4, 2)	Standard	288	15,840	55
(4, 2)	Alternate	288	13,824	48
(3, 4)	Standard	3,456	345,600	100
(3, 4)	Alternate	3,456	237,312	≈ 68.7
(2, 6)	Standard	46,080	6,958,080	151
(2, 6)	Alternate	46,080	3,363,840	73
(1, 8)	Standard	645,120	134,184,960	208
(1, 8)	Alternate	645,120	38,707,200	60

5 A final question

The standard setting used in this article was that X is an N-dimensional projective variety with dim Sing (X) ≤ N − 2 and ℰ is a rank N vector bundle spanned by a vector space V of sections such there is at least one section s ∈ V with at least one isolated solution on X. Then, the total number σ_ℰ of singularities of the solution sets of systems parameterized by a general line in ℙ(V^∗) was computed in Theorem 18. In particular, under modest conditions, σ_ℰ can be computed with a straight-forward formula. Thus one can compare the Bézout number B_ℰ with the number of singularities σ_ℰ by way of the singularity ratio r_ℰ = σ_ℰ/B_ℰ.

Rather than considering a general pencil, suppose that one takes a pencil ℋ = 〈g, f〉 such that g ∈ ℙ(V^∗) is general and f ∈ ℙ(V^∗) is arbitrary. Let σ_ℰ,ℋ,f denote the number of such singularities away from f . Clearly, σ_ℰ,ℋ,f ≤ σ_ℰ, but it would be nice to have a reasonable lower bound for σ_ℰ,ℋ,f to understand how the singularities at f impact the singularities away from f . In practice, these two numbers are relatively close, which, combined with the study in [12], justifies using σ_ℰ as a measure of quality of the homotopy ℋ.

Question. Is there a good upper bound (independent of the choice of f ) for the deficiency ratio

σ E − σ E , H , f B E ?

Thinking (loosely), it might be reasonable to expect that you could lose at most the Bézout number of singularities at f . However, this is false as shown in the following two simple examples of three quadrics on ℙ³. Here,

E = [ 2 ] P 3 ⊕ [ 2 ] P 3 ⊕ [ 2 ] P 3

with B_ℰ = 8 and σ_ℰ = 48. First, when f is the system (xy, xz, yz) giving the coordinate lines and g is general, then σ_ℰ,ℋ,f = 33. Hence σ_ℰ − σ_ℰ,ℋ,f = 48 − 33 = 15 > 8 = B_ℰ with

σ E − σ E , H , f B E = 15 8 > 1 .

Similarly, when f is the system (x² − y, xy − z, xz − y²) giving a twisted cubic and g is general, then σ_ℰ,ℋ,f = 34. Hence σ_ℰ − σ_ℰ,ℋ,f = 48 − 34 = 14 > 8 = B_ℰ with

σ E − σ E , H , f B E = 7 4 > 1 .

Finally, let us give one substantial example. In § 4.3 we gave two counts for general lines with the same multihomogeneous structure as the Alt–Burmester problems. Let g be a general system of the type specified in (26) and let f be a general Alt–Burmester system of type (4, 2) in (ℙ¹)⁴×(ℙ²)⁴, i.e., synthesizing a four-bar linkage with four given poses and two precision points selected generically. In particular, for simplicity, the following shows f in affine coordinates

(27) p j z 1 + q j − x 1 p j ˆ z 1 ˆ + q j ˆ − x 1 ˆ − p 1 z 1 + q 1 − x 1 p 1 ˆ z 1 ˆ + q 1 ˆ − x 1 ˆ for j = 2 , 3 , 4 p j z 2 + q j − x 2 p j ˆ z 2 ˆ + q j ˆ − x 2 ˆ − p 1 z 2 + q 1 − x 2 p 1 ˆ z 2 ˆ + q 1 ˆ − x 2 ˆ for j = 2 , 3 , 4 θ 1 z 1 + q j − x 1 θ 1 ˆ z 1 ˆ + q j ˆ − x 1 ˆ − p 1 z 1 + q 1 − x 1 p 1 ˆ z 1 ˆ + q 1 ˆ − x 1 ˆ for j = 5 , 6 θ 2 z 2 + q j − x 2 θ 2 ˆ z 2 ˆ + q j ˆ − x 2 ˆ − p 1 z 2 + q 1 − x 2 p 1 ˆ z 2 ˆ + q 1 ˆ − x 2 ˆ for j = 5 , 6 θ 1 θ 1 ˆ − 1 θ 2 θ 2 ˆ − 1

where

affine variables on the four ℂ’s are given by θ1,θ1ˆ,θ2,θ2ˆ defining rotations;
the affine variables on the four ℂ²’s are given by X1,Z1,X1ˆ,Z1ˆ,X2,Z2,X2ˆ,Z2ˆ defining the four-bar linkage with pivots xj,xˆj and legs zj,zˆj;
eight parameters pj,pˆj for j = 1, . . . , 4 with pj⋅pjˆ=1 defining the orientations of the four poses; and
twelve parameters qj,qjˆ for j = 1, . . . , 6 defining the points (poses correspond with j = 1, . . . , 4 and precision points correspond with j = 5, 6).

For a general linear homotopy (as computed in Table 2), we have B_ℰ = 288 and σ_ℰ = 15,840. Letting ℋ = 〈g, f〉 and using Bertini, see [3; 4], it was computed in about 15.5 hours with 24 cores that σ_ℰ,ℋ,f = 15,064, i.e. 15,064 singularities for the systems (excluding those of f ) are parameterized by ℋ. Therefore

σ E − σ E , H , f B E = 15 , 840 − 15 , 064 288 = 776 288 = 97 36 > 2.694 .

Hence, it would be interesting to know how large this deficiency ratio can be.

We note that the solution set of f for the “standard” formulation of the general Alt–Burmester problem of type (4, 2) as specified above consists of the following:

four irreducible components of dimension two
1. all at “infinity” and isomorphic to the vanishing of the last two polynomials in (ℙ¹)⁴;
five irreducible components of dimension one
1. one corresponds with the four-bar linkage degenerating to a 2R linkage, i.e., x1=x2,xˆ1=xˆ2,z1=z2,zˆ1=zˆ2,
2. four arising from having one of the pivots xj,xˆj or legs zj,zˆj at “infinity”; and
64 nonsingular isolated solutions
1. 60 “finite” (which solve the (4,2) Alt–Burmester problem as in [7]) and 4 at “infinity.”

The solution set decomposition is the same for a nonempty Zariski open set of the parameter space.

Funding statement: The second author was in part supported by NSF CCF-2331440, Simons Foundation SFM-00005696, and the Robert and Sara Lumpkins Collegiate Professorship. The third author was in part supported by NSF CCF-2331440 and the Robert and Sara Lumpkins Collegiate Professorship. The last author was in part supported in part by the Huisking Foundation, Inc. Collegiate Research Professorship.

Communicated by: M. Joswig

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Received: 2024-12-09

Revised: 2025-06-18

Published Online: 2025-10-28

Published in Print: 2025-10-27

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Keywords for this article

Ramification point; branch point; polynomial system; Lefschetz pencil; homotopy continuation; numerical algebraic geometry

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