let $d$ be a digraph with skew-adjacency matrix $s(d)$‎. ‎the skew‎ ‎energy of $d$ is defined as the sum of the norms of all‎ ‎eigenvalues of $s(d)$‎. ‎two digraphs are said to be skew‎ ‎equienergetic if their skew energies are equal‎. ‎we establish an‎ ‎expression for the characteristic polynomial of the skew‎ ‎adjacency matrix of the join of two digraphs‎, ‎and for the‎ ‎respective skew energ...

Efficient algorithms are presented for factoring polynomials in the skew-polynomial ring F[x; σ], a non-commutative generalization of the usual ring of polynomials F[x], where F is a finite field and σ: F → F is an automorphism (iterated Frobenius map). Applications include fast functional decomposition algorithms for a class of polynomials in F[x] whose decompositions are “wild” and previously...

A skew Laurent polynomial ring S = R[x;α] is reversible if it has a reversing automorphism, that is, an automorphism θ of period 2 that transposes x and x and restricts to an automorphism γ of R with γ = γ. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent...

Assuming sufficiently many terms of an n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In applications, come along structure: for instance, may be zero outside cone, built from Gröbner basis ideal invariant under action finite group. Thus, show how to take advantage this structur...

An extension L/K of skew fields is called a leftpolynomialextension with polynomial generator 0 if it has a left basis of the form 1, i3, ti’, , Bnm’ for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extens...

For every ring R, we construct a ringed space NCSpec(R) and for every ring homomorphism R → S, a morphism of ringed spaces NCSpec(S) → NCSpec(R). We show that this gives a fully faithful contravariant functor from the category of rings to a category of ringed spaces. If R is a commutative ring, we show that NCSpec(R) can be viewed as a natural completion of Spec(R). We then explain how the spac...

We study the complexity of computation of a tropical Schur polynomial Tsλ where λ is a partition, and of a tropical polynomial Tmλ obtained by the tropicalization of the monomial symmetric function mλ. Then Tsλ and Tmλ coincide as tropical functions (so, as convex piece-wise linear functions), while differ as tropical polynomials. We prove the following bounds on the complexity of computing ove...