Exponential sums and singular hypersurfaces

We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $ -adic local systems on affine spaces in characteristic $p> 0$ whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.

A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field. For a polynomial f ( x , y ) f(x,y) over F q {F_q} of total degree n , our algorithm takes at most \[ n 4.89 log 2 n log ⁡ q {n^{4.89}}{\log ^2}n\log q \] operations in F q {F_q} to factor f ( x , y ) f(x,y) completely. This improves a probabilistic factorization algorithm of von zur Gathen and Kaltofen, which takes \[ O ( n 11 log ⁡ n log ⁡ q ) O({n^{11}}\log n\log q) \] operations to factor f ( x , y ) f(x,y) completely over F q {F_q} . The algorithm can be easily generalized to factor multivariate polynomials over finite fields. We shall give two further applications of the idea involved in the algorithm. One is concerned with exponential sums; the other is related to permutational polynomials over finite fields (a conjecture of Chowla and Zassenhaus).

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

Mean value of some exponential sums and applications to Kloosterman sums

In this paper, we prove new ℓp→ℓq bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number rs, k(l) of representations of a positive integer l as a sum of s positive kth powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.

Several classes of exponential sums and three-valued Walsh spectrums over finite fields

In [4], Carlitz determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz’s results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields. ∗franciscastr@gmail.com, Tel: (787)764-0000, X4696, Fax: (787)281-0651 †iverubio@uprrp.edu, Tel: (787)764-0000 (4759), Fax: (787)773-1717 ‡jmanuel jmvg@yahoo.com, Tel: 787-738-2161

Inversive methods provide an attractive alternative to linear methods for pseudorandom number and vector generation. We present the first nontrivial results on the distribution of sequences of digital inversive pseudorandom numbers and inversive pseudorandom vectors in parts of the period. The proofs are based on a new bound for certain exponential sums over finite fields.

On the L-functions associated with certain exponential sums

Let \({\mathcal{O}}\) be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of \({\mathcal{O}}\) a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman’s Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with distinct residue classes. In this paper, we extend Lev’s result by lifting a recent result of ours to the case of the points being bounded away from each other.

Maximal estimates for Weyl sums on [formula omitted

Congruence arithmetic algorithms for polynomial real zero determination

On small zeros of quadratic forms over finite fields