Polynomial degree bounds for matrix semi-invariants

  title={Polynomial degree bounds for matrix semi-invariants},
  author={Harm Derksen and Visu Makam},

Weyl's polarization theorem for matrix semi-invariants in positive characteristic

Consider the ring of invariants $R(n,m)$ for the left-right action of ${\rm SL}_n \times {\rm SL}_n$ on $m$-tuples of $n \times n$ matrices. It was recently proved that the ring $R(n,m)$ is generated

An exponential lower bound for the degrees of invariants of cubic forms and tensor actions

Determinantal hypersurfaces and representations of Coxeter groups

Given a finite generating set $T=\{g_0,\dots, g_n\}$ of a group $G$, and a representation $\rho$ of $G$ on a Hilbert space $V$, we investigate how the geometry of the set $D(T,\rho)=\{ [x_0 : \dots :

Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$,

Algorithms for orbit closure separation for invariants and semi-invariants of matrices

Two group actions on $m$-tuples of $n \times n$ matrices are considered, one of which is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $SL_n $ which gives efficient algorithms to decide if the orbit closures of two points intersect.

Symbolic determinant identity testing (SDIT) is not a null cone problem; and the symmetries of algebraic varieties

  • Visu MakamA. Wigderson
  • Mathematics, Computer Science
    2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
  • 2020
This paper provides a barrier for the attempts of derandomizing SDIT via these algorithms, and suggests a general method for determining the symmetries of general algebraic varieties, an algorithmic problem that was hardly studied and is central to algebraic complexity.

Explicit tensors of border rank at least $2d-2$ in $K^d \otimes K^d \otimes K^d$ in arbitrary characteristic

For tensors in $\mathbb{C}^d \otimes \mathbb{C}^d \otimes \mathbb{C}^d$, Landsberg provides non-trivial equations for tensors of border rank $2d-3$ for $d$ even and $2d-5$ for $d$ odd were found by

The Capacity of Quiver Representations and Brascamp–Lieb Constants

Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations,

Computing the Degree of Determinants via Discrete Convex Optimization on Euclidean Buildings

  • H. Hirai
  • Computer Science, Mathematics
    SIAM J. Appl. Algebra Geom.
  • 2019
A DCA-oriented algorithm (steepest descent algorithm) to compute the degree of determinants, which is understood as a variant of the combinatorial relaxation algorithm, which was developed earlier by Murota for computing thedegree of the (ordinary) determinant.

Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix

It is proved that a weighted version (WMVP) of MVSP can be solved in psuedo polynomial time, provided arithmetic operations on ${\bf F}$ can be done in constant time.



On generating the ring of matrix semi-invariants

The Procesi-Razmyslov-Formanek approach of proving a strong degree bound for generating matrix invariants is guided, and several interesting structural results for the ring of matrix semi-invariants over fields of characteristic $0 are exhibited.

Hilbert series and degree bounds for matrix (semi-)invariants

Non-commutative Edmonds’ problem and matrix semi-invariants

This paper considers the non-commutative version of Edmonds’ problem: compute the rank of T over the free skew field by using an algorithm of Gurvits, and assuming the above bound of sigma for R(n, m) over Q, deciding whether or not T has non-Commutative rank < n over Q can be done deterministically in time polynomial in the input size and $$sigma}$$σ.

Rings of matrix invariants in positive characteristic

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈

Non-commutative arithmetic circuits with division

It is shown how divisions can be eliminated from non-commutative circuits and formulae which compute polynomials, and the non-Commutative version of the "rational function identity testing" problem is addressed.

Reductive groups are geometrically reductive

Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of

A Deterministic Polynomial Time Algorithm for Non-commutative Rational Identity Testing

A deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over Q is invertible or not, which efficiently solves several problems in different areas which had only exponential-time algorithms prior to this work.

Invariant functions on matrices

  • S. Donkin
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1993
The ring of symmetric polynomials in n variables may be interpreted as a ring of characters of the general linear group GL(n) (see e.g. [4], §3·5). We consider here a generalization of symmetric

Lower bounds for non-commutative computation

  • N. Nisan
  • Computer Science, Mathematics
    STOC '91
  • 1991
The question of the power of negation in this model is shown to be closely related to a well known open problem relating communication complexity and rank, and exponential lower bounds for monotone algebraic circuit size are obtained.