# On semiring complexity of Schur polynomials

On semiring complexity of Schur polynomials comput. complex. c Springer International Publishing AG, part of Springer Nature 2018 https://doi.org/10.1007/s00037-018-0169-3 computational complexity ON SEMIRING COMPLEXITY OF SCHUR POLYNOMIALS Sergey Fomin, Dima Grigoriev, Dorian Nogneng, and Eric Schost Abstract. Semiring complexity is the version of arithmetic circuit com- plexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial s (x ,...,x ) λ k labeled by a partition λ =(λ ≥ λ ≥ ··· ) is bounded by O(log(λ )) 1 2 1 provided the number of variables k is ﬁxed. Keywords. Semiring complexity, Schur function, Young tableaux. Subject classiﬁcation. Primary 68Q25, Secondary 05E05. 1. Introduction and main results Let f (x ,...,x ) be a polynomial with nonnegative integer coeﬃ- 1 k cients. As such, f can be computed using addition and multipli- cation only—without subtraction or division. To be more precise, one can build an arithmetic circuit wherein ◦ each gate performs an operation of addition or multiplication; ◦ the inputs are x ,...,x , possibly along with some positive 1 k integer scalars; ◦ the sole output is f (x ,...,x ). 1 k The semiring complexity (or {+, ×}-complexity)of f is the smallest size of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Complexity Springer Journals

# On semiring complexity of Schur polynomials

, Volume OnlineFirst – Jun 4, 2018
22 pages

/lp/springer_journal/on-semiring-complexity-of-schur-polynomials-8X8qRqu9lR
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Computational Mathematics and Numerical Analysis
ISSN
1016-3328
eISSN
1420-8954
D.O.I.
10.1007/s00037-018-0169-3
Publisher site
See Article on Publisher Site

### Abstract

comput. complex. c Springer International Publishing AG, part of Springer Nature 2018 https://doi.org/10.1007/s00037-018-0169-3 computational complexity ON SEMIRING COMPLEXITY OF SCHUR POLYNOMIALS Sergey Fomin, Dima Grigoriev, Dorian Nogneng, and Eric Schost Abstract. Semiring complexity is the version of arithmetic circuit com- plexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial s (x ,...,x ) λ k labeled by a partition λ =(λ ≥ λ ≥ ··· ) is bounded by O(log(λ )) 1 2 1 provided the number of variables k is ﬁxed. Keywords. Semiring complexity, Schur function, Young tableaux. Subject classiﬁcation. Primary 68Q25, Secondary 05E05. 1. Introduction and main results Let f (x ,...,x ) be a polynomial with nonnegative integer coeﬃ- 1 k cients. As such, f can be computed using addition and multipli- cation only—without subtraction or division. To be more precise, one can build an arithmetic circuit wherein ◦ each gate performs an operation of addition or multiplication; ◦ the inputs are x ,...,x , possibly along with some positive 1 k integer scalars; ◦ the sole output is f (x ,...,x ). 1 k The semiring complexity (or {+, ×}-complexity)of f is the smallest size of

### Journal

Computational ComplexitySpringer Journals

Published: Jun 4, 2018

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