Volume 2010, Issue 638

Let denote the ring of polynomials over the finite field 𝕗 q of characteristic p , and write for the additive closure of the set of k th powers of polynomials in . Define G q ( k ) to be the least integer s satisfying the property that every polynomial in of sufficiently large degree admits a strict representation as a sum of s k th powers. We employ a version of the Hardy-Littlewood method involving the use of smooth polynomials in order to establish a bound of the shape G q ( k ) ≦ Ck log k + O ( k log log k ). Here, the coefficient C is equal to 1 when k < p , and C is given explicitly in terms of k and p when k > p , but in any case satisfies C ≦ 4/3. There are associated conclusions for the solubility of diagonal equations over , and for exceptional set estimates in Waring's problem.

In this note an improvement of Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurfaces, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces.

The key result in the paper concerns two transformations, Φ : ( ρ , ψ ) ↦ φ and 𝔹 t : ψ ↦ φ , where ρ , ψ , φ are states on the algebra of non-commutative polynomials, or equivalently joint distributions of d -tuples of non-commuting operators. These transformations are related to free probability: if ⊞ is the free convolution operation, and { ρ t } is a free convolution semigroup, we show that The maps {𝔹 t } were introduced by Belinschi and Nica as a semigroup of transformations such that 𝔹 1 is the bijection between infinitely divisible distributions in Boolean and free probability theories. They showed that for γ t the free heat semigroup and Φ the Boolean version of the Kolmogorov representation for infinitely divisible measures, The more general map Φ[ ρ , ψ ] comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bożejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in [Anshelevich, J. Math.], is closely related to the map Φ. On the other hand, one can describe explicitly the action of all the transformations above on free Meixner families, which provides clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Φ[ ρ , ψ ] and descriptions of its fixed points and range.

Given a submodule and a free resolution of J one can define a certain vector-valued residue current whose annihilator is J . We make a decomposition of the current with respect to Ass J that corresponds to a primary decomposition of J . As a tool we introduce a class of currents that includes usual residue and principal value currents; in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets.

In this paper we continue the study of free holomorphic functions on the noncommutative ball where B (ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ, and n = 1, 2, . . . or n = ∞. These are noncommutative multivariable analogues of the analytic functions on the open unit disc 𝔻 ≔ { z ∈ ℂ : | z | < 1}. The theory of characteristic functions for row contractions (elements in ) is used to determine the group of all free holomorphic automorphisms of [ B (ℋ) n ] 1 . It is shown that , the Moebius group of the open unit ball 𝔹 n ≔ { λ ∈ ℂ n : ∥ λ ∥ 2 < 1}. We show that the noncommutative Poisson transform commutes with the action of the automorphism group . This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra C *( S 1 , . . . , S n ), which leave invariant the noncommutative disc algebra . This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We show that the unitarily implemented automorphisms of the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra , respectively, are determined by the free holomorphic automorphisms of [ B (ℋ) n ] 1 , via the noncommutative Poisson transform. Moreover, we prove that . We also prove that any completely isometric automorphism of the noncommutative disc algebra has the form where . We deduce a similar result for the noncommutative Hardy algebra , which, due to Davidson-Pitts results on the automorphisms of , implies that is isomorphic to the group of contractive automorphisms of . We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group . This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under . Finally, in the last section, we prove a maximum principle for free holomorphic functions on the noncommutative ball [ B (ℋ) n ] 1 and provide some extensions of the classical Schwarz lemma to our noncommutative setting.

We consider parabolic systems , where the coefficient functions possess a potential and linear growth in ∇ u . Assuming a coerciveness and natural entropy condition, but no monotonicity condition, we prove a Morrey estimate for ∇ u which implies C α -regularity in the case of two dimensions.

We derive a cut-and-paste surgery formula of Seiberg–Witten invariants for negative definite plumbed rational homology 3-spheres. It is similar to (and motivated by) Okuma's recursion formula [Trans. Amer. Math. Soc.], 4.5, targeting analytic invariants of splice-quotient singularities. Combining the two formulas automatically provides a proof of the equivariant version [Némethi, Line bundles associated with normal surface singularities, World Sci. Publ., 2007], 5.2(b), of the Seiberg–Witten invariant conjecture [Némethi and Nicolaescu, Geom. Topol. 6: 269–328, 2002] for these singularities.

Minimal log discrepancies (mld's) are related not only to termination of log flips [Shokurov, Algebr. Geom. Metody 246: 328–351, (2004)] but also to the ascending chain condition (ACC) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the ACC conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension ≦ d imply the following: the ACC conjecture for a -lc thresholds, in particular, for canonical and log canonical (lc) thresholds in dimension ≦ d ; the ACC conjecture for lc thresholds in dimension ≦ d + 1; and termination of log flips in dimension ≦ d + 1 for effective lc pairs. In particular, when d = 3 we can drop the assumptions on LMMP and boundedness of canonical Mori-Fano varieties.