In contrast with Tennenbaum’s theorem [8] that says Peano Arithmetic (PA) has no nonstandard computable model, Shepherdson [6] constructed a computable nonstandard model for a very weak fragment of PA, called Open Induction (Iopen) in which the induction scheme is only allowed to be applied for quantifier-free formulas (with parameters). Since then several attempts have been made from both side...

In this work we present the formalization of an algorithm to compute the Echelon Form of a matrix. We have proved its existence over Bezout domains and we have made it executable over Euclidean domains, such as Z and K[x]. This allows us to compute determinants, inverses and characteristic polynomials of matrices. The work is based on the HOL-Multivariate Analysis library, and on both the Gauss...

! $# , where # is the identity matrix. However, only two different types of primeness, ZLP and MLP, have been defined in [13], which correspond to the case % and a polynomial containing &(' % variables . To my knowledge, nothing has been done for the other cases until the work of Oberst [6], surely because the complexity of the matrices increases with the number & . The main contribution of [6]...

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-mult...

Efficient algorithms are derived for computing the entries of the Bezout resultant matrix for two univariate polynomials of degree n and for calculating the entries of the Dixon–Cayley resultant matrix for three bivariate polynomials of bidegree (m, n). Standard methods based on explicit formulas require O(n3) additions and multiplications to compute all the entries of the Bezout resultant matr...

In this short note, by using a general three circles theorem, we show the rigidity of sharp Bezout estimate first found Gang Liu on nonnegatively curved Riemann surfaces.

In this paper, some familiar and new results on arithmetical rings, modules, Bezout rings (not necessarily commutative) are provided. particular, we examine relationships between their localizations by maximal ideals, saturated submodules saturations, localizable properties of annihilators finitely generated modules over diagonalizable with flat right quasi-projective Hermite Pierce stalks, Kru...