research-article

Identity Testing for Radical Expressions

Authors:

Mahsa Shirmohammadi,

James WorrellAuthors Info & Claims

LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

Article No.: 8, Pages 1 - 11

https://doi.org/10.1145/3531130.3533331

Published: 04 August 2022 Publication History

Abstract

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial and nonnegative integers a1, …, ak and d1, …, dk, written in binary, test whether the polynomial vanishes at the real radicals, i.e., test whether . We place the problem in coNP assuming the Generalised Riemann Hypothesis (GRH), improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals. Next we consider a restricted version, called 2-RIT, where the radicals are square roots of prime numbers, written in binary. It was known since the work of Chen and Kao [16] that 2-RIT is at least as hard as the polynomial identity testing problem, however no better upper bound than PSPACE was known prior to our work. We show that 2-RIT is in coRP assuming GRH and in coNP unconditionally. Our proof relies on theorems from algebraic and analytic number theory, such as the Chebotarev density theorem and quadratic reciprocity.

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Cited By

Jindal GGaillard L(2023)On the Order of Power Series and the Sum of Square Roots ProblemProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597079(354-362)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597079

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cover image ACM Conferences

LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

August 2022

817 pages

ISBN:9781450393515

DOI:10.1145/3531130

Editor:
Christel Baier

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Published: 04 August 2022

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LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science

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Cited By

Jindal GGaillard L(2023)On the Order of Power Series and the Sum of Square Roots ProblemProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597079(354-362)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597079

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