Counting isolated points outside the image of a polynomial map
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Boulos El Hilany
Abstract
We consider a generic family of polynomial maps f := (f1, f2) : ℂ2 → ℂ2 with given supports of polynomials, and degree deg f := max(deg f1, deg f2). We show that the (non-) properness of maps f in this family depends uniquely on the pair of supports, and that the set of isolated points in ℂ2 ∖ f(ℂ2) has a size of at most 6 deg f. This improves an existing upper bound (deg f – 1)2 proven by Jelonek. Moreover, for each n ∈ ℕ, we construct a dominant map f as above, with deg f = 2n + 2, and having 2n isolated points in ℂ2 ∖ f(ℂ2). Our proofs are constructive and can be adapted to a method for computing isolated missing points of f. As a byproduct, we describe those points in terms of singularities of the bifurcation set of f.
Acknowledgements
The author is grateful to Zbigniew Jelonek for introducing him to the problem and for his helpful remarks. The author would also like to thank Elias Tsigaridas for fruitful discussions. The author thanks the anonymous referee for their valuable suggestions on earlier versions of the manuscript, for pointing out mistakes therein and mentioning works of Fernando, Gamboa and Ueno on the subject. The author is grateful to the Mathematical Institute of the Polish Academy of Sciences in Warsaw for their financial support and hospitality.
Communicated by: C. Scheiderer
Funding: Part of this work was supported by the Austrian Science Fund (FWF) P33003.
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Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
