We derive polynomial identities of arbitrary degree n for syzygies degrees numerical semigroups $$S_m\!=\!\langle d_1,\ldots ,d_m\rangle $$ and show that $$n\ge m$$ they contain higher genera $$G_r\!=\!\sum _{s\in {\mathbb Z}_>\!\!\setminus S_m}s^ r$$ $$S_m$$ . find a number $$g_m\!=\!B_m-m+1$$ algebraically independent $$G_r$$ equations, related any $$g_m+1$$ genera, where $$B_m\!=\! \sum _{k=...

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action...

This article explains the similar appearance of two polynomial identities involving Dickson polynomials in char. 2, one found by Abhyankar, Cohen and Zieve, other author.

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorith...