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...

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 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...

In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and suucient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of S-polyno...

In recent years various authors have studied C*-algebras generated by singular integral operators, Wiener-Hopf operators, Toeplitz operators, etc., and representations of these algebras. Analysis in these algebras is accomplished modulo a suitable ideal (usually the commutator and/or compact ideal). A similar procedure is used in the theory of pseudo-differential operators except that the empha...

Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...