On the Complexity of Symmetric Polynomials

Bläser, Markus; Jindal, Gorav

doi:10.4230/lipics.itcs.2019.47

Cited by 2 publications

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“…A fairly direct consequence of our techniques is the following theorem which can be considered a partial generalization of the result of Bläser-Jindal [BJ18] for algebraic formulas.…”

Section: Shifted Variants For Formulasmentioning

confidence: 83%

“…We formally state and prove Lemma 1.4 in Section 4.1, followed by its application to the proof of Theorem 1.3 in Section 4.2. We discuss further applications of some of these ideas to extending the result of Bläser and Jindal's [BJ18] in Section 4.3 and to the question of proving lower bounds on the partial derivative complexity of a product of algebraically independent polynomials in Section 4.4. We conclude with some open questions in Section 5.…”

Section: Other Related Work the Algebraic Complexity Of Schur Polynom...mentioning

confidence: 99%

“…Hence, for each b ∈ C n , there exists an a ∈ C n such that e(a) = b. The reason this is relevant to the result of [BJ18] is that if we have an efficient algorithm for 'inverting' e in this way and we additionally have an efficient algorithm for computing f sym , then we immediately obtain an efficient algorithm for computing f E on any given input b ∈ C n by first inverting the map e to obtain an a as above, and then applying the algorithm for f sym to obtain f (a) = f E (b). The main technical result in Bläser and Jindal's work is to show how to invert the map e as above using an algebraic circuit.…”

Section: Introductionmentioning

confidence: 99%

“…Newton iteration and formula complexity. Theorem 1.3 is motivated in part by a recent result of Bläser and Jindal [BJ18] who prove the following interesting result about symmetric polynomials. As mentioned earlier, it is known that any symmetric polynomial…”

Section: Introductionmentioning

confidence: 99%

“…We could hope to use the theorem of Bläser and Jindal to prove that if the Schur polynomial has a small formula, then so does the determinant. However, this doesn't quite work, since the proof of [BJ18] only yields small circuits for the polynomial f E , even if we assume that the polynomial f sym has small formulas.…”

Section: Introductionmentioning

confidence: 99%

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

Chaugule¹,

Kumar²,

Limaye³

et al. 2019

Preprint

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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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“…A fairly direct consequence of our techniques is the following theorem which can be considered a partial generalization of the result of Bläser-Jindal [BJ18] for algebraic formulas.…”

Section: Shifted Variants For Formulasmentioning

confidence: 83%

Section: Other Related Work the Algebraic Complexity Of Schur Polynom...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chaugule¹,

Kumar²,

Limaye³

et al. 2019

Preprint

View full text Add to dashboard Cite

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On the complexity of invariant polynomials under the action of finite reflection groups

Vu¹

2022

Preprint

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Let be a finite reflection group and K[ 1 , . . . , ] be a multivariate polynomial ring over a field K. Let K[ 1 , . . . , ] be a set containing all invariant polynomials under the action of . Then the Chevalley-Shephard-Todd theorem states that there exists a sequence of homogeneous polynomials 1 , . . . , such that for any polynomial in K[ 1 , . . . , ] , there exists a unique polynomial new in K[ 1 , . . . , ], where 1 , . . . , are new variables, such that new ( 1 , . . . , ) = ( 1 , . . . , ). In this paper, we study the arithmetic complexity to compute new knowing and ( 1 , . . . , ). Our algorithm works for any finite reflection group and any set of generators ( 1 , . . . , ). Previously such a result was only known for a symmetric group or a directed product of symmetric groups and 1 , . . . , elementary symmetric functions.

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On the Complexity of Symmetric Polynomials

Cited by 2 publications

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

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

On the complexity of invariant polynomials under the action of finite reflection groups

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