On the complexity of symmetric polynomials

Loading...
Thumbnail Image
Journal Title
Journal ISSN
Volume Title
Conference article in proceedings
Date
2019-01-01
Major/Subject
Mcode
Degree programme
Language
en
Pages
1-14
Series
10th Innovations in Theoretical Computer Science, ITCS 2019, Leibniz International Proceedings in Informatics, LIPIcs, Volume 124
Abstract
The fundamental theorem of symmetric polynomials states that for a symmetric polynomial fSym ∈ C[x1, x2, . . ., xn], there exists a unique “witness” f ∈ C[y1, y2, . . ., yn] such that fSym = f(e1, e2, . . ., en), where the ei’s are the elementary symmetric polynomials. In this paper, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(fSym) of fSym. We show that the arithmetic complexity L(f) of f is bounded by poly(L(fSym), deg(f), n). To the best of our knowledge, prior to this work only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton’s iteration on power series. As a corollary of this result, we show that if VP 6= VNP then there exist symmetric polynomial families which have super-polynomial arithmetic complexity. Furthermore, we study the complexity of testing whether a function is symmetric. For polynomials, this question is equivalent to arithmetic circuit identity testing. In contrast tothis, we show that it is hard for Boolean functions.
Description
| openaire: EC/H2020/759557/EU//ALGOCom
Keywords
Arithmetic circuits, Arithmetic complexity, Elementary symmetric polynomials, Newton’s iteration, Power series, Symmetric polynomials
Other note
Citation
Bläser , M & Jindal , G 2019 , On the complexity of symmetric polynomials . in A Blum (ed.) , 10th Innovations in Theoretical Computer Science, ITCS 2019 . , 47 , Leibniz International Proceedings in Informatics, LIPIcs , vol. 124 , Schloss Dagstuhl - Leibniz-Zentrum für Informatik , pp. 1-14 , Innovations in Theoretical Computer Science Conference , San Diego , California , United States , 10/01/2019 . https://doi.org/10.4230/LIPIcs.ITCS.2019.47