The Complexity of Factors of Multivariate Polynomials

Bürgisser, Peter

doi:10.1007/s10208-002-0059-5

Cited by 33 publications

(21 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer to [12,Prop. 9.3.2] for an equivalent formulation in terms of complexity classes that goes back to [5]. (In particular see [5, per m ∈ Ω m c for infinitely many m. Conjecture 1.2 implies Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

No occurrence obstructions in geometric complexity theory

Bürgisser

Ikenmeyer

Panova

2018

J. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VPws and VNP. Mulmuley and Sohoni [37] suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL n 2 (C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed in [37]. 1.3. Conjecture (Mulmuley and Sohoni 2001). For all c ∈ N ≥1 , for infinitely many m, there exists a partition λ occurring in C[Z m c ,m ] but not in C[Ω m c ].This conjecture implies Conjecture 1.2 by the above reasoning. Conjecture 1.3 on the existence of occurrence obstructions has stimulated a lot of research and has been the main focus of researchers in geometric complexity theory in the past years, see Section 1 (b). Unfortunately, this conjecture is false! This is the main result of this work. More specifically, we show the following.1.4. Theorem. Let n, d, m be positive integers with n ≥ m 25 and λ ⊢ nd. If λ occurs in C[Z n,m ], then λ also occurs in C[Ω n ]. In particular, Conjecture 1.3 is false.One can likely improve the bound on n by a more careful analysis. 1 (b). Related work: Kronecker coefficients. Kronecker coefficients are fundamental quantities that have been the object of study in algebraic combinatorics for a long time [40]. A difficulty in their study is that there is no known counting interpretation of them [42]. The Kronecker coefficient k(λ, µ, ν) of three partitions λ, µ, ν of the same size d is defined as the dimension of the, where [λ] denotes the irreducible S d -module of type λ and S d is the symmetric group on d symbols; see [32, I §7, internal product]. We write n × d for the rectangular partition (d, . . . , d) of size nd and call k n (λ) := k(λ, n × d, n × d) the rectangular Kronecker coefficient of λ ⊢ nd. In [25] it was shown that deciding positivity of Kronecker coefficients in general is NP-hard, but this proof fails for rectangular formats.Let K n (λ) denote the multiplicity by which the irreducible GL n 2 -module of type λ ⊢ nd occurs in C[Ω n ] d . We call the numbers K n (λ) the GCT-coefficients. In [37] it was realized that GCTcoefficients can be upper bounded by rectangular Kronecker coefficients: we have K n (λ) ≤ k n (λ) for λ ⊢ nd. In fact, the multiplicity of λ in the coordinate ring of the orbit GL n 2 · det n equals the so-called symmetric rectangular Kronecker coefficient [12], which is upper bounded by k n (λ).Note that an occurrence obstruction for Z n,m ⊆ Ω n is a partition λ for which K n (λ) = 0 and such that λ occurs in C[Z n,m ]. Since hardly anything was known about the...

show abstract

Section: Introductionmentioning

confidence: 99%

No occurrence obstructions in geometric complexity theory

Bürgisser

Ikenmeyer

Panova

2018

J. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…N P problem, that they call the Geometric Complexity Theory (GCT) program. The starting point is Valiant's conjecture [63] (see also [65,8]) that the permanent hypersurface in m 2 variables (i.e., the set of m × m matrices X with perm m (X) = 0) cannot be realized as an affine linear section of the determinant hypersurface in n(m) 2 variables with n(m) a polynomial function of m. Their program (at least up to [51]) translates the problem of proving Valiant's conjecture to proving a conjecture in representation theory. In this paper we give an exposition of the program outlined in [50,51], present the representation-theoretic conjecture in detail, and present a framework for reducing their representation theory questions to easier questions by taking more geometric information into account.…”

Section: Introductionmentioning

confidence: 99%

An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to $\mathbf{VP}\neq\mathbf{VNP}$

Bürgisser¹,

Landsberg²,

Manivel³

et al. 2011

SIAM J. Comput.

Self Cite

101

View full text Add to dashboard Cite

show abstract

“…It is easy to show that the function field F(ε) here can be equivalently replaced by the ring F[ε, ε −1 ] of Laurent polynomials, or, the field F((ε)) of formal Laurent series (use mod εF[ε]). A reason why these objects appear in algebraic complexity can be found in [11,Section 5.2] and [37,Section 5]. They help algebrize the notion of "infinitesimal approximation" (in real analysis think of ε → 0 & 1/ε → ∞).…”

Section: Our Resultsmentioning

confidence: 99%

Untitled

Guo¹,

Saxena²,

Sinhababu³

2019

Theory of Comput.

View full text Add to dashboard Cite

Testing whether a set f of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. The complexity of algebraic independence testing is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). Previously, the best complexity bound known was NP #P (Mittmann, Saxena, Scheiblechner, Trans. AMS 2014). In this article we put the problem in AM ∩ coAM. In particular, dependence testing is unlikely to be NP-hard. Our proof uses methods of algebraic geometry. We estimate the size of the image and the sizes of the preimages of the polynomial map f over the finite field. A gap between the corresponding sizes for independent and for dependent sets of polynomials is utilized in the AM protocols.

show abstract

The Complexity of Factors of Multivariate Polynomials

Cited by 33 publications

References 46 publications

No occurrence obstructions in geometric complexity theory

No occurrence obstructions in geometric complexity theory

An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to $\mathbf{VP}\neq\mathbf{VNP}$

Untitled

Contact Info

Product

Resources

About