相关论文: Lower Bounds for diophantine Approximation

The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus…

We present a new method for solving symbolically zero--dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and…

An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The…

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…

A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete (FD-) method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schr\"odinger equation…

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…

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…

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be…

A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j…

Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value…

We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the…

Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing…

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…

In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…

In the present work, we present a new discrete logarithm algorithm, in the same vein as in recent works by Joux, using an asymptotically more efficient descent approach. The main result gives a quasi-polynomial heuristic complexity for the…

We present a new algorithm for the computation of the irreducible factors of degree at most $d$, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the…

The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…

Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…