Nathan Ilten, Zach Teitler
We show that the product rank of the $3\times 3$ determinant is $5$, and the product rank of the $3\times 3$ permanent is $4$. As a corollary, we obtain that the tensor ranks of the $3 \times 3$ determinant and permanent are $5$ and $4$, respectively. We show moreover that the border product rank of the $n \times n$ permanent is larger than $n$ for any $n\geq 3$.
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We initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomially related to one with such symmetry, then Valiant's conjecture on permanent v. determinant is true.
Let $D_{m,n}^r$ and $P_{m,n}^r$ denote the subschemes of $\mathbb{P}^{mn-1}$ given by the $r\times r$ determinants (respectively the $r\times r$ permanents) of an $m\times n$ matrix of indeterminates. In this paper, we study the geometry of the Fano schemes $\mathbf{F}_k(D_{m,n}^r)$ and $\mathbf{F}_k(P_{m,n}^r)$ parametrizing the $k$-dimensional planes in $\mathbb{P}^{mn-1}$ lying on $D_{m,n}^r$ and $P_{m,n}^r$, respectively. We prove results characterizing which of these Fano schemes are smooth, irreducible, and connected; and we give examples showing that they need not be reduced. We show th…
We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.
The method of shifted partial derivatives was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent $\ell^{n-m} perm_m$ cannot be realized inside the $GL_{n^2}$-orbit closure of the determinant $ det_n$ when $n>2m^2+2m$. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from Gupta et. al. regarding the shifted partial derivatives of the dete…
Recursive Koszul flattenings of determinant and permanent tensors
本文递归运用 Koszul 展平方法推导行列式与积和式张量的张量秩新下界，实现两类张量按秩完全分离，并确定四阶情形的精确秩。
Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.
We investigate the space of syzygies of the apolar ideals $\det_n^\perp$ and ${\rm perm}_n^\perp$ of the determinant $\det_n$ and permanent ${\rm perm}_n$ polynomials. Shafiei had proved that these ideals are generated by quadrics and provided a minimal generating set. Extending on her work, in characteristic distinct from two, we prove that the space of relations of $\det_n^{\perp}$ is generated by linear relations and we describe a minimal generating set. The linear relations of ${\rm perm}_n^{\perp}$ do not generate all relations, but we provide a minimal generating set of linear and quadra…
We compute Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree in terms of generators and relations.
We present new lower bounds for the symmetric border rank of the n x n determinant for all n. Further lower bounds are given for the 3 x 3 permanent.
Denote by B^k_{2,K} the locus of vector bundles of rank two and canonical determinant. We show that for a generic curve of genus g, B^k_{2,K} is non-empty if g is sufficiently large.
We generalize the prior linked symplectic Grassmannian construction, applying it to to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. We apply this general machinery to prove new results on nonemptiness and dimension of rank-2 Brill-Noether loci in a range of degrees.
We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to H^0(S^2(E))$.
In the 1990's, Bertram, Feinberg and Mukai examined Brill-Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill-Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.
This paper establishes the correctness of a conjecture of Bertram-Feinberg and Mukai for a special class of globally generated rank-two bundles with canonical determinant over a generic Riemann surface of genus at least four.
Consider the scheme B_{2,L}^k of stable vector bundles of rank two and fixed determinant L which have at least k sections. Under suitable numerical conditions and for generic L, we show the existence of a component of the expected dimension of B_{2,L}^k.
The Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum decomposition. We describe some of the symmetries of the decomposition and set-theoretic defining equations for the terms of the decomposition.
We propose a three-step program for the classification of stable rank 2 bundles on the projective space $\mathbb{P}^3$ inspired by an article by Hartshorne and Rao. While this classification program has been successfully completed for stable rank 2 bundles with even determinant and $c_2\le5$, much less is known for bundles with odd determinant. After revising the known facts about these objects, we list all possible spectra and minimal monads for stable rank 2 bundles with odd determinant and $c_2\le8$. We provide a full classification of all bundles with positive minimal monads, provide a neg…
Continuing our previous study of modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank …
The rank of the $5\times 5$ permanent tensor is sixteen
本文确定 5×5 永久张量的张量秩为 16，其下界与已知上界匹配，此前仅知其秩在 15 或 16 之间，研究借助高阶 Koszul 扁平化和 Glynn 公式推进了该问题的结论。
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…
Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P^8 there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result.
We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 \times 3$ permanent is at least $14$.
By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd determinant. For each c_2 \geq 7, we prove that a component of their moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on Gamma having enough sections when twisted by V. Moreover we prove that M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma) along the curve parametrizing lines contained in X.
Classification of monads and moduli components of stable rank 2 bundles with odd determinant and $c_2=10$
本文完整分类了$\mathbb{P}^3$上陈类为$c_1=-1,c_2=10$的稳定秩2向量丛所对应的正极小单子，证明模空间$\mathcal{B}(-1,10)$存在新的不可约分支，并刻画了奇行列式稳定秩2向量丛的谱实现条件。
We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.
see the old abstract and the comments here.
There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition…
Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\geq k$. We show that $B^4(2,K_C)$ it has an irreducible component $\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\geq 8$. Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fits in a exact sequence $0\to\mathcal O_C(D)\to E\to K_C(-D)\to 0$, with $D$ a general effective divisor of degree three, and the corresponding …
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. Mulmuley and Sohoni (SIAM J. Comput., 2001) 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 orbi…
A blowing-up formula for the intersection cohomology of the moduli of rank 2 Higgs bundles over a curve with trivial determinant
本文证明了行列式平凡曲线上秩2 Higgs丛模空间交集上同调的爆破公式，并据此在技术假设下推导出其庞加莱多项式。
The border rank of the $4 \times 4$ determinant tensor is twelve
该研究通过特定方法证明复数域上4×4行列式张量的边界秩至少为12，结合已知上界确定其边界秩为12，明确了该张量的边界秩精确值。
Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix
本文利用 Bernstein-Gelfand-Gelfand 对应产生的谱序列，结合 Gröbner 基与单纯复形方法，给出了 2×n 一般矩阵的 2×2 子永久式理想的极小自由消解，解决了该理想非素且非 Cohen-Macaulay 情形下的结构问题。
Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula
该研究利用Glynn's formula，通过替换Houston–Goucher–Johnston提出的张量分解的部分内容，将n×n行列式张量秩的上界改进为Bₙ−2ⁿ⁻¹+n+1，同时为n≥17时该行列式多项式的Waring秩提供了改进上界。
We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ is trivial.
We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently gen…
The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.
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