Related papers: Polynomial equations modulo prime numbers

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…

We introduce a sequence $P_{2n}$ of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of $P_{2n}$ and its degree $d$ has a limit when…

Consider polynomials $F_1,\dots,F_s$ in $\K[X_1,\dots,X_n]$ over a field $\K$, their zero-set $V(F_1,\dots,F_n)$ in $\Kbar^n$ and its decomposition into equidimensional components $V_0,\dots,V_n$ (with $V_i$ either empty or of dimension $i$…

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for…

Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying…

Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…

Elementary symmetric polynomials $S_n^k$ are used as a benchmark for the bounded-depth arithmetic circuit model of computation. In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$ can be computed with much fewer…

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…

To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like…

In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we…

Let f(x) be a polynomial with integer coefficients, let n be a positive integer, and let p be an odd prime. Then the mapping x-->f(x) sends Z/p^n into Z/p^n. We study the topological structure of this mapping.

We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…

We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in…

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order…

In this paper, we obtain a lower bound for the number of primes $p\leq x$ such that $p-1$ is a sum of two squares and $p+2$ has a bounded number of prime factors. The proof uses the vector sieve framework, involving a semi-linear sieve and…