Ideals Modulo a Prime

The main focus of this paper is on the problem of relating an ideal [Formula: see text] in the polynomial ring [Formula: see text] to a corresponding ideal in [Formula: see text] where [Formula: see text] is a prime number; in other words, the reduction modulo[Formula: see text] of [Formula: see text]. We first define a new notion of [Formula: see text]-good prime for [Formula: see text] which does depends on the term ordering [Formula: see text], but not on the given generators of [Formula: see text]. We relate our notion of [Formula: see text]-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo [Formula: see text] from the term ordering, thus letting us show that all but finitely many primes are good for [Formula: see text]. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

Term Ordering-Based Query Expansion Technique for Hindi-English CLIR System

The quality of retrieval documents in CLIR is often poor compared to IR system due to (1) query mismatching, (2) multiple representations of query terms, and (3) un-translated query terms. The inappropriate translation may lead to poor quality of results. Hence, automated query translation is performed using the back-translation approach for improvement of query translation. This chapter mainly focuses on query expansion (Q.E) and proposes an algorithm to address the drift query issue for Hindi-English CLIR. The system uses FIRE datasets and a set of 50 queries of Hindi language for evaluation. The purpose of a term ordering-based algorithm is to resolve the drift query issue in Q.E. The result shows that the relevancy of Hindi-English CLIR is improved by performing Q.E. using a term ordering-based algorithm. The outcome achieved 60.18% accuracy of results where Q.E has been performed using a term ordering based algorithm, whereas the result of Q.E without a term ordering-based algorithm stands at 57.46%.

Computing all border bases for ideals of points

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

Automated Invention of Strategies and Term Orderings for Vampire

In this work we significantly increase the performance of the Vampire and E automated theorem provers (ATPs) on a set of loop theory problems. This is done by developing EmpireTune, an AI system that automatically invents targeted search strategies for Vampire and E. EmpireTune extends previous strategy invention systems in several ways. We have developed support for the Vampire prover, further extended Vampire by new mechanisms for specifying term orderings, and EmpireTune can now automatically invent suitable term ordering for classes of problems. We describe the motivation behind these additions, their implementation in Vampire and EmpireTune, and evaluate the systems with very good results on the AIM (loop theory) ATP benchmark.

Gröbner bases for syzygy modules of border bases

Given an order ideal 𝒪 and an 𝒪-border basis of a 0-dimensional polynomial ideal, it was shown by Huibregtse that the liftings of the neighbor syzygies (i.e. of the fundamental syzygies of neighboring border terms) form a system of generators for the syzygy module of the border basis. We elaborate on Huibregtse's proof and transform it into explicit algorithmic form. Based on this, we are able to exhibit explicit conditions on a module term ordering τ such that the liftings of the neighbor syzygies are in fact a τ-Gröbner basis. Finally, we construct term orderings satisfying these conditions in an explicit algorithmic way.