相关论文: Lower Bounds for diophantine Approximation
De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting…
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…
We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of…
In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division…
We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem…
We give an approximate algorithm of computing holonomic systems of linear differential equations for definite integrals with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping…
A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity…
We study an important special case of the differential elimination problem: given a polynomial parametric dynamical system $\mathbf{x}' = \mathbf{g}(\boldsymbol{\mu}, \mathbf{x})$ and a polynomial observation function $y =…
We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…
The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
We consider a version of geometric programming problem consisting in minimizing a function given by the maximum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption,…
For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible…
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system…
Correspondence is a ubiquitous problem in computer vision and graph matching has been a natural way to formalize correspondence as an optimization problem. Recently, graph matching solvers have included higher-order terms representing…
The realization space of geometric constraint systems is given by the vanishing locus of polynomials corresponding to natural geometric constraints. Such geometric constraint systems arise in many real-world scenarios such as structural…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the…