Related papers: Verifying Buchberger's Algorithm in Reduction Ring…

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix $G$ with the Smith forms of an irreducible polynomial system matrix $P$ giving rise…

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…

The aim of this paper is to present an explicit reduction algorithm for Hilbert modular groups over arbitrary totally real number fields. An implementation of the algorithm is available to download from [19]. The exposition is…

For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness.…

We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of…

We present algorithms for computing the reduced Gr\"{o}bner basis of the vanishing ideal of a finite set of points in a frame of ideal interpolation. Ideal interpolation is defined by a linear projector whose kernel is a polynomial ideal.…

The complexity of Gr\"{o}bner computations has inspired many improvements to Buchberger's algorithm over the years. Looking for further insights into the algorithm's performance, we offer a threaded implementation of classical Buchberger's…

Cryptographic primitives are fundamental for information security: they are used as basic components for cryptographic protocols or public-key cryptosystems. In many cases, their security proofs consist in showing that they are reducible to…

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program…

Gr\"obner bases are a fundamental tool when studying ideals in multivariate polynomial rings. More recently there has been a growing interest in transferring techniques from the field case to other coefficient rings, most notably Euclidean…

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gr\"obner bases. We present several linear algebra algorithms for computing Gr\"obner…

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known…

We show that time complexity analysis of higher-order functional programs can be effectively reduced to an arguably simpler (although computationally equivalent) verification problem, namely checking first-order inequalities for validity.…

Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably…

As machine learning is increasingly used in essential systems, it is important to reduce or eliminate the incidence of serious bugs. A growing body of research has developed machine learning algorithms with formal guarantees about…

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gr\"obner bases for the ideal of the data points.…

Designing quantum processors is a complex task that demands advanced verification methods to ensure their correct functionality. However, traditional methods of comprehensively verifying quantum devices, such as quantum process tomography,…

A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to…