An efficient algorithm is presented for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the ideal is given by its Gröbner basis, no extra Gröbner basis computation is needed for factoring a polynomial over the extension field. We will only use linear algebra to get a polynomial over the base field by a gener...

We call a module  essentially retractable if HomR for all essential submodules N of M. For a right FBN ring R, it is shown that: (i)  A non-zero module  is retractable (in the sense that HomR for all non-zero ) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii)  A non-zero module  is essentially retractable if and on...

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid actio...

The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the pro...

A bstract We investigate the reduction of Feynman integrals to master using Gröbner bases in a rational double-shift algebra Y which integration-by-parts (IBP) relations form left ideal. The problem reducing given family can then be solved once and for all by computing basis ideal formed IBP relations. demonstrate this explicitly several examples. introduce so-called first-order normal-form we ...

Let S be a semigroup. In certain cases we give some characterizations of extreme amenability of S and we show that in these cases extreme left amenability and extreme right amenability of S are equivalent. Also when S is a compact topological semigroup, we characterize extremely left amenable subalgebras of C(S), where C(S) is the space of all continuous bounded real valued functions on S

Let K be a number field and q an integral ideal in OK. A result of Tatuzawa [11] from 1973, computes the asymptotic (with error term) for ideals with norm at most x class narrow ray group modulo q. This bounds term constant whose dependence on is explicit but not explicit. The aim this paper to prove fully bound term.

This paper is about Hilbert modular forms on certain Hilbert modular varieties associated with a totally real field L. Let p be unramified in L. We reduce to the inert case and consider modular forms modulo p. We study the ideal of modular forms with q-expansion equal to zero modulo p, find canonical elements in it, and obtain as a corollary the congruences for the values of the zeta function o...