Related papers: On Affine Tropical F5 Algorithms
Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of…
In this paper, a polynomial-time algorithm is given to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Groebner basis of the Z[x]-module generated by the column vectors of F. The algorithm…
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 introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a…
We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We…
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…
This short note is the generalization of Faugere F4-algorithm for polynomial rings with coefficients in Euclidean rings. This algorithm computes successively a Groebner basis replacing the reduction of one single s-polynomial in…
A generalized criterion for signature-based algorithms to compute Gr\"obner bases is proposed in this paper. This criterion is named by "generalized criterion", because it can be specialized to almost all existing criteria for…
We use tropical algebras as platforms for a very efficient digital signature protocol. Security relies on computational hardness of factoring one-variable tropical polynomials; this problem is known to be NP-hard.
Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton.…
We present an elegant, generic and extensive formalization of Gr\"obner bases in Isabelle/HOL. The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria…
A generalized criterion for signature related algorithms to compute Gr\"obner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gr\"obner basis, including the famous F5 algorithm,…
Gr\"obner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field…
Given a finite set of closed rational points of affine space over a field, we give a Gr\"obner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the…
This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.
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…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
The homogeneous spectrum of a multigraded finitely generated algebra (in the sense of Brenner-Schr\"oer) always admits an embedding into a toric variety that is not necessarily separated, a so-called toric prevariety. In order to have a…
In this work we develop the theory of Gr\"obner bases for modules over the ring of univariate linearized polynomials with coefficients from a finite field.
Let k be a finite field of characteristic p>0. We construct a theory of weights for overholonomic complexes of arithmetic D-modules with Frobenius structure on varieties over k. The notion of weight behave like Deligne's one in the l-adic…