Related papers: Determinants, Choices and Combinatorics
We present a general formula for the Atiyah-Sutcliffe determinant function, which holds for any integer $n \geq 2$, as a global factor times a sum of terms, with each term similar to a higher degree cross-ratio. The formula is to our…
We use the elimination theory to explicitly construct the (n-3)! order polynomial in one of the variables of the scattering equations. The answer can be given either in terms of a determinant of Sylvester type of dimension (n-3)! or a…
We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.
Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and Sutcliffe stated…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
We generalize the Atiyah problem on configurations and the related Atiyah--Sutcliffe conjectures 1 and 2 using finite graphs, configurations of points and tensors. Our conjectures are intriguing geometric inequalities, defined using the…
For the case of 4 points in Euclidean space, we present a computer aided proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah's determinant along with an elegant factorization of the square of the imaginary part of…
We relate the n! conjecture (by Garsia and Haiman) to the geometry of principal nilpotent pairs, and state a conjecture generalizing the n! conjecture to arbitrary semisimple algebraic groups. We also show, using Borel's fixed point…
From any configuration of finitely many points in Euclidean three-space, Atiyah constructed a determinant and conjectured that it was always non-zero. Atiyah and Sutcliffe (hep-th/0105179) amass a great deal of evidence it its favour. In…
In this short note, we show that the Atiyah-Sutcliffe conjectures for $n = 2m$, related to the unitary groups $U(2m)$, imply the author's analogous conjectures, which are associated with the symplectic groups $Sp(m)$. The proof is based on…
Rota's Basis Conjecture is a well known problem from matroid theory, that states that for any collection of $n$ bases in a rank $n$ matroid, it is possible to decompose all the elements into $n$ disjoint rainbow bases. Here an asymptotic…
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix.…
Extending Sellers' result, Das et al. recently proved some congruence results for generalized overcubic partitions using theta functions and posed some related conjectures. In this paper, we provide a combinatorial proof of a result in…
We provide a short proof of a conic version of the colorful Carath\'eodory theorem for oriented matroids. Holmsen's extension of the colorful Carath\'eodory theorem to oriented matroids (Advances in Mathematics, 2016) already encompasses…
We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
This paper presents a proof of the monodromy conjecture for determinantal varieties. Our strategy centers on an in-depth analysis of monodromy zeta functions, leveraging a generalized A'Campo formula, an examination of multiple contact…
This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple…
For any three $\,n\times n\,$ matrices $\,A,B,X\,$ over a commutative ring $\,S$, we prove that $\,{\rm det}\,(A+B-AXB)={\rm det}\,(A+B-BXA) \in S$. This apparently new formula may be regarded as a ``ternary generalization'' of Sylvester's…
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…