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Fournier, Hervé; Limaye, Nutan; Mahajan, Meena; Srinivasan, Srikanth

doi:10.4086/toc.2017.v013a009

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…The shifted partial derivative complexity of elementary symmetric polynomials is studied by Fournier et al [7], where strong lower bounds are proved, which in turn give complexity lower bounds for depth-four circuits.…”

Section: Related Workmentioning

confidence: 99%

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Gesmundo¹,

Landsberg²

2019

Theory of Comput.

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We answer a question of K. Mulmuley. Efremenko et al. (Math. Comp., 2018) have shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.

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Section: Related Workmentioning

confidence: 99%

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Gesmundo¹,

Landsberg²

2019

Theory of Comput.

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“…2 The same idea also yields small constant-depth formulas for the complete homogeneous symmetric polynomials. Symmetric polynomials have also been used to prove lower bounds for several interesting models of computation including homogeneous and inhomogeneous ΣΠΣ formulas [NW97,SW01,Shp01], homogeneous multilinear formulas [HY11] and homogenous ΣΠΣΠ formulas [FLMS17]. Further, via reductions, the elementary and power symmetric polynomials have been used to define restricted models of algebraic computation known as the symmetric circuit model [Shp01] and the Σ ∧ Σ model (or Waring rank), which in turn have been significantly investigated (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

Schur Polynomials do not have small formulas if the Determinant doesn't!

Chaugule¹,

Kumar²,

Limaye³

et al. 2019

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Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [HRTS00, OW15] and Geometric complexity theory [IP17].However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas.As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.

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