Related papers: New Facts about Berezinians
A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping…
In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose…
New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally - using…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
Subresultant is a powerful tool for developing various algorithms in computer algebra. Subresultants for polynomials in standard basis (i.e., power basis) have been well studied so far. With the popularity of basis-preserving algorithms,…
In this paper, we define a braid invariant, the purified determinant $P(b)$ of a braid $b$, considering the determinant of the crossing matrix of a pure braid derived from $b$, and show that $P(b_1 b_2)=P(b_2 b_1)$ for any pair of…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified…
This work presents closed formulas for determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.
In the supergeometric setting, the classical identification between differential forms of top degree and volume elements for integration breaks down. To address this, generalized notions of differential forms were introduced:…
We study the classical invariant theory of the Bezoutiant R(A,B) of a pair of binary forms A,B. It is shown that R(A,B) admits a Taylor expansion whose coefficients are (essentially) the odd transvectants (A,B)_{2r+1}. Moreover, R(A,B) is…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type…
We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein's inequality and give a formulation of this last result…
We found another N=1 odd superanalog of complex structure (the even one is widely used in the theory of super Riemann surfaces). New N=1 superconformal-like transformations are similar to anti-holomorphic ones of nonsupersymmetric complex…
A (global) determinantal representation of hypersurface in P^n is a matrix, whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…