相关论文: Bounds on the realizations of zero-nonzero pattern…

To determine the dimension of null space of any given linearized polynomial is one of vital problems in finite field theory, with concern to design of modern symmetric cryptosystems. But, the known general theory for this task is much far…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…

In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials…

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-\xi}< \epsilon$ for a generic ternary form $Q$ is $\abs{x}\ll \epsilon^{-1}$. We also establish…

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant…

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

Since the breakthrough superpolynomial multilinear formula lower bounds of Raz (Theory of Computing 2006), proving such lower bounds against multilinear algebraic branching programs (mABPs) has been a longstanding open problem in algebraic…

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemer\'{e}di theorem for certain families of polynomials. Namely, we show that if $P_1, \dots, P_m \in…

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings…

Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $\tau$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar…

Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some…

We give new explicit constructions of several fundamental objects in linear-algebraic pseudorandomness and combinatorics, including lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over finite fields. Our focus…