相关论文: Lower bound for ranks of invariant forms

We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of…

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where…

We describe the variety of fixed points of a unipotent operator acting on the space of matrices. We compute the determinant and the rank of a generic (symmetric, or anti-symmetric) matrix in the fixed variety, yielding information about the…

We discuss an approach to the secant non-defectivity of the varieties parametrizing $k$-th powers of forms of degree $d$. It employs a Terracini type argument along with certain degeneration arguments, some of which are based on toric…

We give lower bounds for genera of components of fiber products of holomorphic maps between compact Riemann surfaces, extending results on genera of components of algebraic curves of the form $A(x)-B(y)=0,$ where $A$ and $B$ are rational…

In this paper, we give upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and characterize the equality cases. These bounds theoretically improve and generalize some known results of Duan et al.[X. Duan, B.…

The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. This…

Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for…

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they…

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by \[ {\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}. \] This result significantly improves…

Classical invariant theory establishes a systematic correspondence between algebraic and smooth invariants for compact and reductive Lie groups. However, the extension of these results to non-compact and non-reductive regimes remains a…

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carath\'eodory spaces and we describe a concrete method to lower bound the…

A lower bound for the interleaving distance on persistence vector spaces is given in terms of rank invariants. This offers an alternative proof of the stability of rank invariants.

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show…

The canonical dimension is an invariant attached to admissible representations of p-adic reductive groups, which has only received significant attention in the case of mod-p representations. In the case of complex representations, the…

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…

We find higher rank generalizations of the Razumov--Stroganov sum rules at $q=-e^{i\pi\over k+1}$ for $A_{k-1}$ models with open boundaries, by constructing polynomial solutions of level one boundary quantum Knizhnik--Zamolodchikov…

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the…

We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices…