eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-08-11
48:1
48:21
10.4230/LIPIcs.APPROX-RANDOM.2017.48
article
On Some Computations on Sparse Polynomials
Volkovich, Ilya
In arithmetic circuit complexity the standard operations are +,x. Yet, in some scenarios exponentiation gates are considered as well. In this paper we study the question of efficiently evaluating a polynomial given an oracle access to its power. Among applications, we show that:
* A reconstruction algorithm for a circuit class c can be extended to handle f^e for f in C.
* There exists an efficient deterministic algorithm for factoring sparse multiquadratic polynomials.
* There is a deterministic algorithm for testing a factorization of sparse polynomials, with constant individual degrees, into sparse irreducible factors. That is, testing if f = g_1 x ... x g_m when f has constant individual degrees and g_i-s are irreducible.
* There is a deterministic reconstruction algorithm for multilinear depth-4 circuits with two multiplication gates.
* There exists an efficient deterministic algorithm for testing whether two powers of sparse polynomials are equal. That is, f^d = g^e when f and g are sparse.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol081-approx-random2017/LIPIcs.APPROX-RANDOM.2017.48/LIPIcs.APPROX-RANDOM.2017.48.pdf
Derandomization
Arithmetic Circuits
Reconstruction