License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.48
URN: urn:nbn:de:0030-drops-75976
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7597/
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Volkovich, Ilya

On Some Computations on Sparse Polynomials

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Abstract

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.

BibTeX - Entry

@InProceedings{volkovich:LIPIcs:2017:7597,
  author =	{Ilya Volkovich},
  title =	{{On Some Computations on Sparse Polynomials}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{48:1--48:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7597},
  URN =		{urn:nbn:de:0030-drops-75976},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.48},
  annote =	{Keywords: Derandomization, Arithmetic Circuits, Reconstruction}
}

Keywords: Derandomization, Arithmetic Circuits, Reconstruction
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Issue Date: 2017
Date of publication: 11.08.2017

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