• Corpus ID: 238744137

Equations for GL invariant families of polynomials

@inproceedings{Breiding2021EquationsFG,
  title={Equations for GL invariant families of polynomials},
  author={Paul Breiding and Christian Ikenmeyer and Mateusz Michałek and Reuven Hodges},
  year={2021}
}
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from Question 7 in [32] and Problem 13 in [34], which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and… 

References

SHOWING 1-10 OF 37 REFERENCES
Equations for Lower Bounds on Border Rank
TLDR
It is shown that these newly obtained polynomials do not vanish on the matrix multiplication operator M 2, which gives a new proof that the border rank of the multiplication of 2×2 matrices is seven.
No Occurrence Obstructions in Geometric Complexity Theory
TLDR
The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP ws and VNP and it is proved that the approach to separating these orbit closures by exhibiting occurrence obstructions is impossible.
Even partitions in plethysms
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 GL d ( C ) -representation of highest weight 2λ
Symmetrizing tableaux and the 5th case of the Foulkes conjecture
TLDR
A complete representation theoretic decomposition of the vanishing ideal of the 5th Chow variety in degree 5 is obtained, it is shown that there are no degree 5 equations for the 6th Chow varieties, and some representation theoretics degree 6 equations are found.
Solving polynomial systems via homotopy continuation and monodromy
TLDR
Under the theoretical assumption that monodromy actions are generated uniformly, it is shown that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions.
Toward a Salmon Conjecture
TLDR
Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3×3×4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9, furnishing a computational solution of an open problem in algebraic statistics.
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the The denominator is taking on this, book interested. This book for
On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions
TLDR
A slight generalization of Hermite's reciprocity theorem is proved, which proves Foulkes' conjecture for a new infinite family of cases.
3× 3 MINORS OF CATALECTICANTS
Secant varieties of Veronese embeddings of projective space are classical varieties whose equations are far from being understood. Minors of catalecticant matrices furnish some of their equations,
The Computational Complexity of Plethysm Coefficients
TLDR
This work is the first to apply techniques from discrete tomography to the study of plethysm coefficients and derive new lower and upper bounds and in special cases even combinatorial descriptions for plethYSm coefficients, which the authors consider to be of independent interest.
...
1
2
3
4
...