On the Generalized Weights of a Class of Trace Codes

We build a class of codes using hermitian forms and the functional trace code. Then we give a general expression of the rth minimum distance of our code and compute general bounds for the weight hierarchy by using exponential sums. We also get the minimum distance and calculate the rth generalized Hamming weight d_r in some special cases.     

On the distribution of k-free integers

We give two new bounds for double exponential sums of type II, by which we get new results (under RH, as before) for the distribution of k-free integers for k ≧ 5.     

Exponential sums of nonlinear congruential pseudorandom number generators with Rédei functions

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. We give new bounds of exponential sums with sequences of iterations of Rédei functions over prime finite fields, which are much stronger than bounds known for general nonlinear congruential pseudorandom number generators.     

Improvement to Moreno–Morenoʼs theorems

In this work, we introduce the p-weight degree of a polynomial over a finite field with respect to a subset of the variables. Using this p-weight, we improve the results of Moreno and Moreno for polynomial equations and for exponential sums over finite fields. We prove that our results cannot be improved in general because a family of polynomials where our bounds are attained is provided. Combining our result with a result of Cao and Sun, we give a p-adic improvement to the p-divisibility of general diagonal equations.     

Exponential sums and singular hypersurfaces

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

Twisted exponential sums of polynomials in one variable

The twisted T-adic exponential sums associated to a polynomial in one variable are studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums. This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.     

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x ₁ + s)... (x _ν + s) ≡ (y ₁ + s)... (x _ν + s) ? 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.     

On the multiplicative properties modulo m of numbers with missing digits

We obtain upper bounds for character sums modulo a composite number over sets of numbers with missing digits in a number system. We derive results on the solvability of congruences of the form x ₁ ? x _t ≡ λ (mod m) in the numbers with missing digits and also asymptotic formulas for the number of solutions.     

Ramanujan’s 1 ψ 1 summation, Hecke-type double sums, and Appell–Lerch sums

We use a specialization of Ramanujan??s ₁ ?? ₁ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell?CLerch sums and theta functions.     

Multiplicative bounds for L 1-norms of exponential sums

This paper is a sequel to the author's papers [1–8] devoted to lower multiplicative bounds on L ₁ norms and their applications. Now we give estimates for L ₁ norms of exponential sums and prove the result announced in [8]. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Exponential sums with Dickson polynomials

We give new bounds of exponential sums with sequences of iterations of Dickson polynomials over prime finite fields. This result is motivated by possible applications to polynomial generators of pseudorandom numbers.     

Bounds for norms of the matrix inverse and the smallest singular value

In the first part, we obtain two easily calculable lower bounds for ‖A^-1‖, where ‖·‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖A^-1‖_∞ and ‖A^-1‖₁ in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖A^-1‖₁ in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.     

On Lower Bounds for the L 1-Norm of Exponential Sums

In the paper a method for finding lower bounds of the L ₁-norm of some exponential sums is described.     

Coloring,sparseness and girth

Using the classical analysis resolution of singularities algorithm of [G4], we generalize the theorems of [G3] on Rⁿ sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x), such as a polynomial with integer coefficients, is divisible by various powers of a prime p when x is an integer. Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron.     

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.