Abstract

ABSTRACT We describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm of Caruso, Roe, and Vaccon for calculating the eigenvalues and eigenvectors of a p-adic matrix.

Highlights

  • Let k be a field and let f1, . . . , fm ∈ k[x1, . . . , xn] be polynomials such that the variety defined by f1, . . . , fm has dimension 0

  • A significant bottleneck in computing the exact solutions of such a polynomial system is the complexity of the field extension required to write down all of the solutions

  • A popular method to compute the solutions exactly is to compute a triangular decomposition for the ideal f1, . . . , fm and solve for the coordinates via backsubstitution

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Summary

INTRODUCTION

Let k be a field and let f1, . . . , fm ∈ k[x1, . . . , xn] be polynomials such that the variety defined by f1, . . . , fm has dimension 0. Our idea to improve on the algorithm of [CRV17] is to adapt the ideas from classical numerical linear algebra to the pnumerical setting; to use an iterative scheme to compute the eigenvectors or eigenvalues. To our knowledge, this is the first appearance of a p-adic numerical algorithm based on iterating matrix multiplication to solve for eigenvalues and eigenvectors. Our algorithms are available as Julia packages, and can be obtained from: https://github.com/a-kulkarn/pAdicSolver https://github.com/a-kulkarn/Dory (For technical reasons related to the Julia package system, we choose to divide our implementation between two packages.)

Precision, norms, and condition numbers
Design choices for the implementation
Matrix factorizations
The eigenvector problem: semantics
Commentary on the classical algorithm
The power iteration algorithm
The Schur form algorithm
18: Update B and V
POLYNOMIAL SYSTEM SOLVING
Findings
The algorithm

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