Identity Testing for Radical Expressions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912073887252480 |
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| author | Balaji, Nikhil Nosan, Klara Shirmohammadi, Mahsa Worrell, James |
| author_facet | Balaji, Nikhil Nosan, Klara Shirmohammadi, Mahsa Worrell, James |
| contents | We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial $f\in \mathbb{Z}[x_1, \ldots, x_k]$ and nonnegative integers $a_1, \ldots, a_k$ and $d_1, \ldots,$ $d_k$, written in binary, test whether the polynomial vanishes at the real radicals $\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}$, i.e., test whether $f(\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}) = 0$. 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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_07961 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Identity Testing for Radical Expressions Balaji, Nikhil Nosan, Klara Shirmohammadi, Mahsa Worrell, James Computational Complexity Logic in Computer Science Symbolic Computation We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial $f\in \mathbb{Z}[x_1, \ldots, x_k]$ and nonnegative integers $a_1, \ldots, a_k$ and $d_1, \ldots,$ $d_k$, written in binary, test whether the polynomial vanishes at the real radicals $\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}$, i.e., test whether $f(\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}) = 0$. 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 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. |
| title | Identity Testing for Radical Expressions |
| topic | Computational Complexity Logic in Computer Science Symbolic Computation |
| url | https://arxiv.org/abs/2202.07961 |