Smooth Herzog projective curves
We prove that smooth projective curves admitting a squarefree Groebner degeneration have genus 0.
math.AC↗
arXiv subjects
Hang Huang
Publications and source records attributed to Hang Huang.
Boundary Distribution Estimation for Precise Object Detection
In the field of state-of-the-art object detection, the task of object localization is typically accomplished through a dedicated subnet that emphasizes bounding box regression. This subnet traditionally predicts the object's position by regressing the box's center position and scaling factors. Despite the widespread adoption of this approach, we have observed that the localization results often suffer from defects, leading to unsatisfactory detector performance. In this paper, we address the shortcomings of previous methods through theoretical analysis and experimental verification and present an innovative solution for precise object detection. Instead of solely focusing on the object's center and size, our approach enhances the accuracy of bounding box localization by refining the box edges based on the estimated distribution at the object's boundary. Experimental results demonstrate the potential and generalizability of our proposed method.
cs.CV↗
On Linear spaces of of matrices bounded rank
Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has been a longstanding problem to classify spaces of bounded rank four. Before our study, no non-classical example of such a space was known. We exhibit two non-classical examples of such spaces and give the full classification of basic spaces of bounded rank four. There are exactly four such up to isomorphism. We also take steps to bring together the methods of the linear algebra community and the algebraic geometry community used to study spaces of bounded rank.
math.AG↗
Geometry of backflow transformation ansatz for quantum many-body fermionic wavefunctions
Wave function ansatz based on the backflow transformation are widely used to parametrize anti-symmetric multivariable functions for many-body quantum problems. We study the geometric aspects of such ansatz, in particular we show that in general totally antisymmetric polynomials cannot be efficiently represented by backflow transformation ansatz at least in the category of polynomials. In fact, one needs a linear combination of at least $O(N^{3N-3})$ determinants to represent a generic totally antisymmetric polynomial. Our proof is based on bounding the dimension of the source of the ansatz from above and bounding the dimension of the target from below.
math-ph↗
Relations between the 2x2 minors of a generic matrix
We prove the case t = 2 of a conjecture of Bruns-Conca-Varbaro, describing the minimal relations between the t x t minors of a generic matrix. Interpreting these relations as polynomial functors, and applying transpose duality as in the work of Sam-Snowden, this problem is equivalent to understanding the relations satisfied by t x t generalized permanents. Our proof follows by combining Koszul homology calculations on the minors side, with a study of subspace varieties on the permanents side, and with the Kempf-Weyman technique (on both sides).
math.AC↗
The InvariantRing package for Macaulay2
We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.
math.AC↗
Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $ω$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$ permanent, and could potentially be used to show $ω=2$. We prove the negative result for complexity theory that its border rank is $16$, resolving a longstanding problem. Regarding its $q=4$ skew cousin in $ C^5\otimes C^5\otimes C^5$, which could potentially be used to prove $ω\leq 2.11$, we show the border rank of its Kronecker square is at most $42$, a remarkable sub-multiplicativity result, as the square of its border rank is $64$. We also determine moduli spaces $\underline{VSP}$ for the small Coppersmith-Winograd tensors.
math.AG↗
Minimal Set of Generators of Ideals Defining Nilpotent Orbit Closures
Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete answer to it.
math.AG↗
Syzygies of Determinantal Thickenings
Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix entries. It is proven by Raicu and Weyman that for a $\mathbf{GL}$-invariant ideal $I \subseteq S$, the linear strands of its minimal free resolution translates via the BGG correspondence to modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. When $I=I_λ$ is the ideal generated by the $\mathbf{GL}$-orbit of a highest weight vector of weight $λ$, they gave a conjectural description of the classes of these $\mathfrak{gl}(m|n)$-modules in the Grothendieck group. We prove their conjecture here. We also give a algorithmic description of how to get the classes of these $\mathfrak{gl}(m|n)$-modules for any $\mathbf{GL}$-invariant ideal $I \subseteq S$.
math.AC↗
Vanishing Hessian, wild forms and their border VSP
Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree $d\geq 3$ as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert schemes.
math.AG↗
Computing Schur complexes
We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud.
math.AC↗
Equations Of Kalman Varieties
The Kalman variety of a linear subspace is a vector space consisting of all endomorphisms that have an eigenvector in that subspace. We resolve a conjecture of Ottaviani and Sturmfels and give the minimal defining equations of the Kalman variety over a field of characteristic 0.
math.AC↗