Skip to Main content Skip to Navigation
Theses

Représentations des polynômes, algorithmes et bornes inférieures

Abstract : Computational complexity is the study of the resources — time, memory, …— needed to algorithmically solve a problem. Within these settings, algebraic complexity theory is the study of the computational complexity of problems of algebraic nature, concerning polynomials. In this thesis, we study several aspects of algebraic complexity. On the one hand, we are interested in the expressiveness of the determinants of matrices as representations of polynomials in Valiant's model of complexity. We show that symmetric matrices have the same expressiveness as the ordinary matrices as soon as the characteristic of the underlying field in different from two, but that this is not the case anymore in characteristic two. We also build the smallest known representation of the permanent by a determinant.On the other hand, we study the computational complexity of algebraic problems. We show that the detection of roots in a system of n homogeneous polynomials in n variables in NP-hard. In line with the “VP = VNP ?”question, which is the algebraic version of “P = NP?” we obtain a lower bound for the computation of the permanent of a matrix by an arithmetic circuit, and we point out the links between this problem and the polynomial identity testing problem. Finally, we give efficient algorithms for the factorization of lacunary bivariate polynomials.
Document type :
Theses
Complete list of metadata

Cited literature [99 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00770148
Contributor : Abes Star :  Contact
Submitted on : Friday, January 4, 2013 - 3:28:15 PM
Last modification on : Wednesday, November 20, 2019 - 2:56:15 AM
Long-term archiving on: : Friday, April 5, 2013 - 5:56:16 AM

File

GRENET_Bruno_2012_These.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-00770148, version 1

Collections

Citation

Bruno Grenet. Représentations des polynômes, algorithmes et bornes inférieures. Autre [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2012. Français. ⟨NNT : 2012ENSL0769⟩. ⟨tel-00770148⟩

Share

Metrics

Record views

872

Files downloads

4257