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- W3038017452 abstract "In this paper we show that the black-box polynomial identity testing for noncommutative polynomials $finmathbb{F}langle z_1,z_2,cdots,z_n rangle$ of degree $D$ and sparsity $t$, can be done in randomized $poly(n,log t,log D)$ time. As a consequence, if the black-box contains a circuit $C$ of size $s$ computing $finmathbb{F}langle z_1,z_2,cdots,z_n rangle$ which has at most $t$ non-zero monomials, then the identity testing can be done by a randomized algorithm with running time polynomial in $s$ and $n$ and $log t$. This makes significant progress on a question that has been open for over ten years. The earlier result by Bogdanov and Wee [BW05], using the classical Amitsur-Levitski theorem, gives a randomized polynomial-time algorithm only for circuits of polynomially bounded syntactic degree. In our result, we place no restriction on the degree of the circuit. Our algorithm is based on automata-theoretic ideas introduced in [AMS08,AM08]. In those papers, the main idea was to construct deterministic finite automata that isolate a single monomial from the set of nonzero monomials of a polynomial $f$ in $mathbb{F}langle z_1,z_2,cdots,z_n rangle$. In the present paper, since we need to deal with exponential degree monomials, we carry out a different kind of monomial isolation using nondeterministic automata." @default.
- W3038017452 created "2020-07-02" @default.
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- W3038017452 date "2016-01-01" @default.
- W3038017452 modified "2023-09-24" @default.
- W3038017452 title "Randomized Polynomial Time Identity Testing for Noncommutative Circuits." @default.
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