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
Summary
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.)
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