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Dima Grigoriev

Publications and source records attributed to Dima Grigoriev.

At least 19 recordsLinked to original sources
arXiv2026

Min-plus Bruhat decomposition

We study min-plus Bruhat cells $B_{trop}^{(n)} \odot p\odot B_{trop}^{(n)}$, where $B_{trop}^{(n)}$ denotes the set of all upper-triangular matrices over $\RR \cup \{\infty\}$, and $p$ is a permutation. It is proved that $\dim (B_{trop}^{(n)} \odot p \odot B_{trop}^{(n)})= \frac{n(n+1)}{2} + l(p)$, where $l(p)$ is the length of $p$ in Coxeter generators. Also we provide an explicit description of the cell $B_{trop}^{(n)}(\RR) \odot p_0\odot B_{trop}^{(n)}(\RR)$ (up to a subset of dimension less than $n^2$) for the highest in Bruhat order permutation $p_0$. Finally, it is shown that for a tropically non-singular matrix $A$ for at most one permutation $p_A$ it can hold $A\in B_{trop}^{(n)}\odot p_A\odot B_{trop}^{(n)}$.

math.AG↗
arXiv2025

Effective resolution of singularities

Consider a projective variety $X \subset \mathbb{P}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E \subset \mathbb{P}^n$, where the degrees of both $X$ and $E$ are at most $d$. We show there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities $(X',E')$, where $(X',E')$ can be embedded in $\mathbb{P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\mathbb{P}^{n'}$.

math.AG↗
arXiv2025

Valuations, bijections, and bases

The aim of this paper is to build a theory of commutative and noncommutative {\it injective} valuations of various algebras (including algebras with zero divisors). The targets of our valuations are (well-)ordered commutative and noncommutative (partial and entire) semigroups including any sub-semigroups of the free monoid $F_n$ on $n$ generators and various quotients. When the range of a valuation of an algebra $A$ is a finitely generated (partial) semigroup, we construct a generalization of the standard monomial bases in $A$, which seems to be new in noncommutative case. Quite remarkably, for any pair of well-ordered valuations one has a canonical bijection between the valuation semigroups, which serves as an analog of the celebrated Jordan-Hölder correspondences and these bijections are ``almost" homomorphisms of the involved semigroups. A spectacular demonstration of this remarkable property of JH-bijections for quantum Schubert cells $A=U_q(w)$ results in mysterious "symplectomorphisms" of involved skew symmetric forms.

math.RA↗
arXiv2025

Degree of 1-dimensional tropical prevariety

For a tropical prevariety $V\subset \RR^n$ (being a finite union of rational polyhedra) we define a tropical Hilbert function $TH_V(k)$ to be the maximal number of tropically independent on $V$ among tropical monomials with degrees at most $k$. In case $\dim V=1$ we define the tropical degree as $$degT (V):=\lim_{k\to \infty} \frac{TH_V(k)}{k}$$ \noindent and prove existence of the limit. We calculate explicitly (a modification of) the tropical degree of $V$ when $V=\cup_l V_l$ is a star, i.e. the union of rays $V_l$ with a common apex.

math.AG↗
arXiv2025

Minimal solutions of tropical linear differential systems

We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential equation in a single unknown has either a solution or a solution at infinity. For a generic system of $n$ tropical linear differential equations in $n$ unknowns, upper and lower bounds on the number of minimal solutions are established. The upper bound involves inversions of a family of permutations which generalize inversions of a single permutation. For $n=1, 2$, we show that the bounds are sharp.

math.AG↗
arXiv2024

Identifying Markov chain models from time-to-event data: an algebraic approach

Many biological and medical questions can be modeled using time-to-event data in finite-state Markov chains, with the phase-type distribution describing intervals between events. We solve the inverse problem: given a phase-type distribution, can we identify the transition rate parameters of the underlying Markov chain? For a specific class of solvable Markov models, we show this problem has a unique solution up to finite symmetry transformations, and we outline a recursive method for computing symbolic solutions for these models across any number of states. Using the Thomas decomposition technique from computer algebra, we further provide symbolic solutions for any model. Interestingly, different models with the same state count but distinct transition graphs can yield identical phase-type distributions. To distinguish among these, we propose additional properties beyond just the time to the next event. We demonstrate the method's applicability by inferring transcriptional regulation models from single-cell transcription imaging data.

math.DS↗
arXiv2024

A tropical version of Hilbert polynomial (in dimension one)

For a tropical univariate polynomial $f$ we define its tropical Hilbert function as the dimension of a tropical linear prevariety of solutions of the tropical Macauley matrix of the polynomial up to a (growing) degree. We show that the tropical Hilbert function equals (for sufficiently large degrees) a sum of a linear function and a periodic function with an integer period. The leading coefficient of the linear function coincides with the tropical entropy of $f$. Also we establish sharp bounds on the tropical entropy.

math.AG↗
arXiv2024

A criterion of containment for tropical hypersurfaces

For tropical $n$-variable polynomials $f, g$ a criterion of containment for tropical hypersurfaces $Trop(f)\subset Trop(g)$ is provided in terms of their Newton polyhedra $N(f), N(g)\subset \mathbb{R}^{n+1}$. Namely, $Trop(f)\subset Trop(g)$ iff for every vertex $v$ of $N(g)$ there exist a homothety $t\cdot N(f), t>0$ and a parallel shift $s:\mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ such that $v\in s(t\cdot N(f))\subset N(g)$.

math.AG↗
arXiv2024

Tropical cryptography III: digital signatures

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.

cs.CR↗
arXiv2023

BASS: Boolean Automorphisms Signature Scheme

We offer a digital signature scheme using Boolean automorphisms of a multivariate polynomial algebra over integers. Verification part of this scheme is based on the approximation of the number of zeros of a multivariate Boolean function.

cs.CR↗
arXiv2021

Tropical Newton-Puiseux polynomials II

Tropical Newton-Puiseux polynomials defined as piece-wise linear functions with rational coefficients at the variables, play a role of tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials being the tropical zeroes of a univariate tropical polynomial with parametric coefficients.

math.AG↗
arXiv2020

Entropy of radical ideal of a tropical prevariety

The entropy of a tropical ideal is introduced. The radical of a tropical ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with vanishing coefficients equals zero. Also we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical tropical ideal having a positive entropy is exhibited.

math.AG↗
arXiv2020

Efficiently and Effectively Recognizing Toricity of Steady State Varieties

We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gröbner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers.

q-bio.MN↗
arXiv2020

Tropical recurrent sequences

Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequences satisfying a given vector with integer coordinates. Finally, we introduce a tropical entropy of a vector and provide some bounds on it.

math.AG↗
arXiv2020

RSA and redactable blockchains

A blockchain is redactable if a private key holder (e.g. a central authority) can change any single block without violating integrity of the whole blockchain, but no other party can do that. In this paper, we offer a simple method of constructing redactable blockchains inspired by the ideas underlying the well-known RSA encryption scheme. Notably, our method can be used in conjunction with any reasonable hash function that is used to build a blockchain. Public immutability of a blockchain in our construction is based on the computational hardness of the RSA problem and not on properties of the underlying hash function. Corruption resistance is based on the computational hardness of the discrete logarithm problem.

cs.CR↗