Related papers: Efficiently Detecting Torsion Points and Subtori

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus…

Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the…

We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP…

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called…

We present a deterministic 2^O(t)q^{(t-2)(t-1)+o(1)} algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree <q, has a root in F_q. A corollary of our method --- the first with complexity sub-linear…

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $\Gamma_1,\ldots,\Gamma_j$ of low-degree…

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global…

We prove that for any Borel probability measure $\mu$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x…

We prove that if $P(X) \in \mathbb{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disk $|z|…

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in…

Let $X$ be a smooth irreducible quasi-projective algebraic variety over a number field $K$. Suppose $X$ is equipped with a $p$-adic \'{e}tale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on…

Let $\mathcal{N} \neq \{0\}$ be a fixed set of integers, closed under multiplication, closed under negation, or containing $\{\pm 1\}$. We prove that any zero of a polynomial in $\mathbf{Z}[X]$ whose coefficients lie in $\mathcal{N}$ can be…

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques…

We show that, for a system of univariate polynomials given in sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine…

We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in…

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show…

Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…