Related papers: A Tropical F5 algorithm
Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Faugere's F4 is a well-known Grobner bases algorithm to solve this problem. However, a serious drawback exists in the Grobner bases based…
Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…
Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that…
Phylogenetic trees provide a fundamental representation of evolutionary relationships, yet the combinatorial explosion of possible tree topologies renders inference computationally challenging. Classical approaches to characterizing tree…
Consider the problem of efficiently evaluating isogenies $\phi: E \to E/H$ of elliptic curves over a finite field $\mathbb{F}_q$, where the kernel $H = \langle G\rangle$ is a cyclic group of odd (prime) order: given $E$, $G$, and a point…
Solving zero-dimensional polynomial systems using Gr\"obner bases is usually done by, first, computing a Gr\"obner basis for the degree reverse lexicographic order, and next computing the lexicographic Gr\"obner basis with a change of order…
We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
Let $A$ be an abelian variety of dimension $g$ together with a principal polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of…
We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The…
Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted…
We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of…
The concepts of tropical-semiring and tropical hypersurface, are extended for an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties…
Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of…
In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications…
We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by…
Can algebraic geometry enhance the sharpness, robustness, and interpretability of modern neural reasoning models by equipping them with a mathematically grounded inductive bias? To answer this, we introduce Tropical Attention, an attention…
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…