We use results from the theory of algebras with polynomial identities (PI algebras) to study the witness complexity of matrix identities. A matrix identity of d× d matrices over a field F is a non-commutative polynomial f(x1, . . . , xn) over F, such that f vanishes on every d×d matrix assignment to its variables. For any field F of characteristic 0, any d > 2 and any finite basis of d× d matri...

For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x1; τ1, δ1] · · · [xn; τn, δn] agrees with the PI degree of R[x1; τ1] · · · [xn; τn] when each (τi, δi) satisfies a qi-skew relation for qi ∈ k and extends to a higher qi-skew τi-derivation. We confirm the quantum Gel’fand-Kirillov conjecture for various quantized coord...

Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On other hand, polynomials related to these were also introduced, for example Szeged polynomial, edgeSzeged PI etc. In this paper, we introduce a concept general polynomial connected strength-weighted graph. It turns out that includes all above mentioned infinitely ma...

The notions of nil, nilpotent or PI-rings (= rings satisfying a polynomial identity) play an important role in the ring theory (see e.g. [8], [11], [20]). Banach algebras with these properties have been studied considerably less and the existing results are scattered in literature. The only exception is the work of Krupnik [13], where the Gelfand theory of Banach PI-algebras is presented. Howev...

Consider a semi-algebraic set A in R constructed from the sets which are determined by inequalities pi(x) > 0, pi(x) ≥ 0, or pi(x) = 0 for a given list of polynomials p1, . . . , pm. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a...

The paper deals with a semi-algebraic set A in R constructed by the inequalities pi(x) > 0, pi(x) ≥ 0, and pi(x) = 0 for a given list of polynomials p1, . . . , pm, and presents several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a certain multi...

In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0, . . . , Ps ≥ 0, where Pi ∈ R[X1, . . . , Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 2 + (k − s) + 1 2 · min{s+1,k−i}

The k-DISJOINT PATHS problem, which takes as input a graph G and k pairs of specified vertices (si, ti), asks whether G contains k mutually vertexdisjoint paths Pi such that Pi connects si and ti, for i = 1, . . . , k. We study a natural variant of this problem, where the vertices of Pi must belong to a specified vertex subset Ui for i = 1, . . . , k. In contrast to the original problem, which ...

In the Knapsack problem we are given a knapsack capacity B, and set N of n items. Each item i has a given size si ≥ 0 and a profit pi ≥ 0. Given a subset of the items A ⊆ N , we define two functions, s(A) = ∑ i∈A si and p(A) = ∑ i∈A pi, representing the total size and profit of the group of items respectively. The goal is to choose a subset of the items, A, such that s(A) ≤ B and p(A) is maximi...