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A motivation to study Gr\"{o}bner theory for fields with valuations comes from tropical geometry, for example, they can be used to compute tropicalization of varieties \citep{maclagan2009introduction}. The computational aspect of this…
A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction.…
In this paper we generalize the classical Groebner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely…
We report on our experiences exploring state of the art Groebner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature…
In a former paper it has been shown that the elliptic Gau{\ss} sums, whose use has been proposed in the context of counting points on elliptic curves and primality tests, can be computed by using modular functions. In this work we give…
Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the…
In this work we study five Grovers algorithm modifications, where each iteration is constructed by two generalized Householder reflections, against inaccuracies in the phases. By using semi-empirical methods, we investigate various…
This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…
An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when…
This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for…
We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…
The Gr\"obner basis detection (GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner basis. This problem was shown to be NP-hard…
Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper…
We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(alpha_1,...,alpha_n). Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over Q can be modified to work…
Computing Gr\"obner bases is known to have a very high upper bound on computation time with respect to input length. Due to the connection between polyhedral geometry and Gr\"obner bases through the Gr\"obner fan, one can attempt an…
Gibbs random fields (GRF) are polymorphous statistical models that can be used to analyse different types of dependence, in particular for spatially correlated data. However, when those models are faced with the challenge of selecting a…
Prime-based ordering which is proved to be admissible, is the encoding of indeterminates in power-products with prime numbers and ordering them by using the natural number order. Using Eiffel, four versions of Buchberger's improved…
In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods…
Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective…
We generalize the Gr\"obner basis method for free D-modules to the case of several term orderings induced by a partition of the set of basic variables. Using this generalized Gr\"obner basis technique we prove the existence and give a…