Low degree polynomial equations: arithmetic, geometry and topology

Let $\cp:=(P_1,...,P_s)$ be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family $\cp$ is in a suitable sense \emph{generic}. We constru...

This paper collects open problems that were presented at the ``Hausdorff Geometry of Polynomials" workshop held on July 10-14, 2023 in Sofia, Bulgaria.

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional poly...

Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the Disquisitiones deals with quadratic residues and nonresidues. Beginning with these fundamental contrib...

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within c...

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals generated by small algebraic integers. This strategy leads to an algorithm for computing the class group whose complexity is possibly as low as $L_{|\Delta_{\mathbf K}|}\left(\frac{1}{3}\right)$.

Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $\mathbf{x} \in [-B,B]^n$ such that $f(\mathbf{x})$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=y^r$ is solvable for $y \in \mathbb{Z}$; then to a polynomial sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=g(y)$ is so...

This paper surveys some applications of moduli theory to issues concerning the distribution of rational points on algebraic varieties. It will appear on the proceedings of the Fano Conference.

Lectures notes in universal algebraic geometry for beginners