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Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program, arXiv eprint math.AG/1004.4802
, 2010
"... of hypersurfaces of degree d in CN that have dual variety of dimension at most k. We apply these equations to the MulmuleySohoni variety GLn2 · [detn] ⊂ P(S n C n2), showing it is an irreducible component of the variety of hypersurfaces of degree n in Cn 2 with dual of dimension at most 2n − 2. We ..."
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Cited by 19 (7 self)
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of hypersurfaces of degree d in CN that have dual variety of dimension at most k. We apply these equations to the MulmuleySohoni variety GLn2 · [detn] ⊂ P(S n C n2), showing it is an irreducible component of the variety of hypersurfaces of degree n in Cn 2 with dual of dimension at most 2n − 2. We establish additional geometric properties of the MulmuleySohoni variety and prove a quadratic lower bound for the determinental bordercomplexity of the permanent. 1.
The stability of the Kronecker products of Schur functions
, 2009
"... ABSTRACT. In the late 1930’s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part o ..."
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Cited by 15 (3 self)
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ABSTRACT. In the late 1930’s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.
ON RECTANGULAR KRONECKER COEFFICIENTS
, 907
"... Abstract. We show that rectangular Kronecker coefficients stabilize when the lengths of the sides of the rectangle grow, and we give an explicit formula for the limit values in terms of invariants of sln. 1. ..."
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Abstract. We show that rectangular Kronecker coefficients stabilize when the lengths of the sides of the rectangle grow, and we give an explicit formula for the limit values in terms of invariants of sln. 1.
Geometric Complexity Theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noether’s Normalization Lemma
"... It is shown that blackbox derandomization of polynomial identity testing (PIT) is essentially equivalent to derandomization of Noether’s Normalization Lemma for explicit algebraic varieties, the problem that lies at the heart of the foundational classification problem of algebraic geometry. Specif ..."
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Cited by 9 (0 self)
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It is shown that blackbox derandomization of polynomial identity testing (PIT) is essentially equivalent to derandomization of Noether’s Normalization Lemma for explicit algebraic varieties, the problem that lies at the heart of the foundational classification problem of algebraic geometry. Specifically: (1) It is shown that in characteristic zero blackbox derandomization of the symbolic trace identity testing (STIT) brings the problem of derandomizing Noether’s Normalization Lemma for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices from EXPSPACE (where it is currently) to P. Next it is shown that assuming the Generalized Riemann Hypothesis (GRH), instead of the blackbox derandomization hypothesis, brings the problem from EXPSPACE to quasiPH, instead of P. Thus blackbox derandomization
On P vs. NP and Geometric Complexity Theory Dedicated to Sri Ramakrishna
, 2011
"... This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and ..."
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Cited by 8 (0 self)
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This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and permanent vs. determinant problems, (2) the Flip Theorem, which formalizes the self referential paradox in the P vs. NP problem, and (3) the Decomposition Theorem, which decomposes the arithmetic P vs. NP and permanent vs. determinant problems into subproblems without self referential difficulty, consisting of positivity hypotheses in algebraic geometry and representation theory and easier hardness hypotheses. 1
FEWNOMIAL SYSTEMS WITH MANY ROOTS, AND AN ADELIC TAU CONJECTURE
"... Abstract. Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, ..."
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Cited by 7 (2 self)
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Abstract. Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the ShubSmale τConjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.
A diagrammatic approach to Kronecker squares
"... Abstract. In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition ν of d there is a polynomial kν with rational coefficients in variables xC, where C runs over the set of isomorphism classes of connected skew diagrams of size at ..."
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Abstract. In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition ν of d there is a polynomial kν with rational coefficients in variables xC, where C runs over the set of isomorphism classes of connected skew diagrams of size at most d, such that for all partitions λ of n, the Kronecker coefficient g(λ, λ, (n − d, ν)) is obtained from kν(xC) substituting each xC by the number of λremovable diagrams in C. We present two applications. First we show that for ρk = (k, k − 1,..., 2, 1) and any partition ν of size d there is a piecewise polynomial function sν such that g(ρk, ρk, (ρk  − d, ν)) = sν(k) for all k and that there is an interval of the form [c,∞) in which sν is polynomial of degree d with leading coefficient the number of standard Young tableaux of shape ν. The second application is new stability property for Kronecker coefficients. Résumé. Dans ce papier nous améliorons une méthode de RobinsonTaulbee pour calculer les coefficients de Kronecker et montrons que pour toute partition ν de d il y a un polynôme kν avec coefficients rationels dans les variables xC, ou C est dans l’ensemble de classes d’isomorphisme des diagrammes gauches connexes de taille non plus que d, tel que pour toute partition λ de n, le coefficient de Kronecker g(λ, λ, (n−d, ν)) est obtenu de kν(xC) en substituant chaque xC pour le nombre de diagrammes λremovables en C. Nous presentons deux applications. Premièrement nous montrons que pour ρk = (k, k − 1,..., 2, 1) et une partition ν de taille d il y a une fonction polynôme par morceaux sν tel que pour toute k on a g(ρk, ρk, (ρk  − d, ν)) = sν(k), et que il y a une interval de la forme [c,∞) dans lequelle sν est polynôme de degre ́ d avec coefficient principal le nombre de tableaux de Young standard de forme ν. La seconde application est une nouveau propriete ́ d’estabilite ́ des coefficients de Kronecker.
On P vs. NP and Geometric Complexity Theory
, 2011
"... This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and ..."
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Cited by 7 (1 self)
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This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and permanent vs. determinant problems, (2) the Flip Theorem, which formalizes the self referential paradox in the P vs. NP problem, and (3) the Decomposition Theorem, which decomposes the arithmetic P vs. NP and permanent vs. determinant problems into subproblems without self referential difficulty, consisting of positivity hypotheses in algebraic geometry and representation theory and easier hardness hypotheses.
Even partitions in plethysms
 J.Algebra
"... Abstract. We prove that for all natural numbers k, n, d with k ≤ d and every partition λ of size kn with at most k parts there exists an irreducible GLd(C)representation of highest weight 2λ in the plethysm Sym k (Sym 2n C d). This gives an affirmative answer to a conjecture by Weintraub (J. Algebr ..."
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Cited by 6 (4 self)
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Abstract. We prove that for all natural numbers k, n, d with k ≤ d and every partition λ of size kn with at most k parts there exists an irreducible GLd(C)representation of highest weight 2λ in the plethysm Sym k (Sym 2n C d). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1): 103114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory. 1.