Related papers: A Proof that the Maximal Rank for Plane Quartics i…

Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.

Erd\H{o}s and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erd\H{o}s distinct distances problem. We consider the analogous problem for…

Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$.

Plane triangulations with all vertices of degree $3$ or $6$ are enumerated. A plane triangulation is said to be akempic if it has a $4$-colouring such that no two adjacent triangles have the same three colours and this colouring is not…

We prove that the dimension of a quartic symmetroid singular along a quadric of codimension 1 is at most 4, if it is not a cone. In the maximal case, the quadric is reducible and consists of rank-3-points. If the quadric is irreducible, it…

We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…

We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations…

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…

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve…

We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the…

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which…

For a 4th order 3-dimensional symmetric tensor with its some entries $1$ or $-1$, we show the analytic sufficient and necessary conditions of its positive definiteness. By applying these conclusions, several strict inequalities is bulit for…

Let F and G be homogeneous polynomials in disjoint sets of variables. We prove that the Waring rank is additive, thus proving the symmetric Strassen conjecture, when either F or G is a power, or F and G have two variables, or either F or G…

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k…

In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39]…

The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…

We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we…