A new family of polynomial identities for computing determinants

Abstract

We give new definitions for the determinant over commutative ring K, noncommutative ring K, noncommutative ring K with associative powers, over noncommutative nonassociative ring K, and study their properties.Let K be a commutative ring, K a noncommutative associative ring, K a noncommutative ring with associative powers (one-monomial associativity), and K be a noncommutative nonassociative ring; let each ring be with division by integers.Here we obtain a new family of polynomial identities for determinant over the ring K, which allows us to give new definitions for determinants over rings K, K, K, and to study their properties. These definitions are closely related to the definition of symmetrized Barvinok's determinant sdet(A) [2] different from the well-known determinant of Dieudonné over a division ring, the quasideterminant [6], and other well-known determinants over noncommutative associative rings [1]. We also estimate the computational complexity of the obtained formulas for the determinants.Let A = (a ij ) be an n×n matrix with elements from the ring K. Let S n be the set of all permutations σ = (σ (1) , . . . , σ (n)) of the set {1, . . . , n}, τ (σ) be the number of inversions in σ. Let S

Permanents as formulas of summation over an algebra with a unique n-ary operation

“…where A 0 is any n × n matrix over the field R. It is of interest to obtain similar results for Schur fuctions, the mixed discriminants, the resultants, and many other planar and space matrix functions over different algebraic systems of various type and its applications (see [4], [7] - [10] and many others). In our view, an answer to the following question is particularly interesting: for which operations f (x 1 , x 2 , .…”

“…Ryser, 1963), which gives the fastest algorithm for computing P er(A). All computation formulas for е-permanents obtained here by means of the known polarization theorem for recovering a polyadditive function from its values on a diagonal [3,4]. (e) If the algebra Ψ 0 contains the unit e, and I n is the identity matrix then the equality eP er f (A) = e for the e-permanent over Ψ 0 is valid.…”

The Determinants over Associative Rings: a Definition, Properties, New Formulas and a Computational Complexity

“…Let S (e) n and S (o) n be the subsets of even and odd permutations in S n , respectively. We call the sequence of elements of a 1σ (1) , . .…”

“…If the ring Q is commutative then the validity of the formula (7) has been established by the author in [2]. If Q = K then the formula (7) for edet(A) has been mentioned (without proof) in [3]. Its validity directly follows from several simple statements of Lemmas 3-5.…”

“…As the symmetrization operator (4) is a symmetric and polyadditive function of its variables, then by means of the polarization formula for each diagonal product a 1σ (1) . .…”