Consider a projective variety X subset of 1n (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor E subset of 1n, 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 1n ' and both X ' and E ' have degrees at most d ' in 1n '.

We introduce and study minimal (with respect to inclusion) solutions of finite 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 the same number of 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.

For tropical n-variable polynomials f, g a criterion of containment for tropical hypersurfaces Trop(f) subset of Trop(g) is provided in terms of their Newton polyhedra N(f), N(g) subset of Rn+1.Namely, Trop(f) subset of Trop(g) iff for every vertex v of N(g) there exists a unique vertex w of N(f) such that for the tangent cones it holds v-w +N(f)w subset of N(g)v. Relying on this criterion an algorithm is designed which tests whether Trop(f) subset of Trop(g) within polynomial complexity. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Propositional proof complexity deals with the lengths of polynomial-time verifiable proofs for Boolean tautologies. An abundance of proof systems is known, including algebraic and semialgebraic systems, which work with polynomial equations and inequalities, respectively. The most basic algebraic proof system is based on Hilbert's Nullstellensatz [7]. Tropical ("min-plus") arithmetic has many applications in various areas of mathematics. The operations are the real addition (as the tropical multiplication) and the minimum (as the tropical addition). Recently, [8, 17, 21] demonstrated a version of Nullstellensatz in the tropical setting. In this paper we introduce (semi)algebraic proof systems that use min-plus arithmetic. For the dual-variable encoding of Boolean variables (two tropical variables x and (x) over bar per one Boolean variable x) and {0, 1}-encoding of the truth values, we prove that a static (Nullstellensatz-based) tropical proof system polynomially simulates daglike resolution and also has short proofs for the propositional pigeon-hole principle. Its dynamic version strengthened by an additional derivation rule (a tropical analogue of resolution by linear inequality) is equivalent to the system Res(LP) (aka R(LP)), which derives nonnegative linear combinations of linear inequalities; this latter system is known to polynomially simulate Krajicek's Res(CP) (aka R(CP)) with unary coefficients. Therefore, tropical proof systems give a finer hierarchy of proof systems below Res(LP) for which we still do not have exponential lower bounds. While the "driving force" in Res(LP) is resolution by linear inequalities, dynamic tropical systems are driven solely by the transitivity of the order, and static tropical proof systems are based on reasoning about differences between the input linear functions. For the truth values encoded by {0,infinity}, dynamic tropical proofs are equivalent to Res(infinity), which is a small-depth Frege system called also DNF resolution. Finally, we provide a lower bound on the size of derivations of a much simplified tropical version of the BINARY VALUE PRINCIPLE in a static tropical proof system. Also, we establish the non-deducibility of the tropical resolution rule in this system and discuss axioms for Boolean logic that do not use dual variables. In this extended abstract, full proofs are omitted.

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.

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.

We offer two very transparent digital signature schemes: one using non-square matrices and the other using scrap automorphisms. The former can be easily converted to a public key encryption scheme.

For a tropical prevariety $V\subset \RR^n$ (being a finite union of rational polyhedra) two tropical polynomials belong to the same congruence class iff they coincide on $V$. We define a tropical Hilbert function $TH_V(k)$ as the dimension of the family of congruence classes of tropical polynomials with tropical degree at most $k$. It is shown that $TH_V(k)$ coincides with a polynomial of degree $\dim V$ for sufficiently big $k$ when $V$ is a polyhedron. For an arbitrary tropical prevariety we formulate it as a conjecture. The conjecture is justified in case $\dim V =0$.

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.

Exposure of cells to non‐optimal growth conditions or to any environment that reduces cell viability can be considered as a stress. In this paper, we are going to highlight the main factors that determine the danger of stress to a cell considered as a biochemical system. To this end, we introduce a new mathematical concept of biosystem stability, where we take into account a signal transduction by deep gene networks. Using this concept and known results on approximations by deep networks, we find asymptotic estimates of the size and the depth of gene regulation networks that define the stress response. We propose a new algorithm to find the gene network approximating a prescribed output. It allows us, with the help of Kolmogorov ‐entropy and the deep neural network theory, to estimate the number of genes involved in regulation of responses on a stress (for example, a heat shock). We show that the main factors that increase the sensitivity of the systems with respect to a stress are the number of biochemical network parameters affected by the stress and sensitivities of kinetic rates with respect to these parameters.

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.

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

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.

This manuscript presents an algorithmic approach to cooperation in biological systems, drawing on fundamental ideas from statistical mechanics and probability theory. Fisher’s geometric model of adaptation suggests that the evolution of organisms well adapted to multiple constraints comes at a significant complexity cost. By utilizing combinatorial models of fitness, we demonstrate that the probability of adapting to all constraints decreases exponentially with the number of constraints, thereby generalizing Fisher’s result. Our main focus is understanding how cooperation can overcome this adaptivity barrier. Through these combinatorial models, we demonstrate that when an organism needs to adapt to a multitude of environmental variables, division of labor emerges as the only viable evolutionary strategy.

Excitable media are prevalent models for describing interesting effects in physical, chemical, and biological systems such as pattern formation, chaos, and wave propagation. In this manuscript, we propose a spatially extended variant of the FitzHugh–Nagumo model that exhibits new effects. In this excitable medium, waves of new kinds propagate. We show that the time evolution of the medium state at the wavefronts is determined by complicated attractors which can be chaotic. The dimension of these attractors can be large and we can control the attractor structure by initial data and a few parameters. These waves are capable transfer complicated information given by a Turing machine or associative memory. We show that these waves are capable to perform cell differentiation creating complicated patterns.

In this paper, we consider reaction-diffusion systems, which describe the propagation of waves with chaotic and time periodic fronts. Using this property, we show that there exist reaction-diffusion models with a few of reagents, which, by a variation of initial data, is capable to generate all possible one-dimensional cell patterns. We describe algorithms, which allow to obtain any prescribed target cell patterns by chaotic waves. Our model can be considered as a reaction-diffusion analogue of universal Turing machine. So, we propose a new robust mechanism of positional information transfer, which, in contrast to Wolpert' gradients, can work at long distances. Universality of our model helps to explain why genes, responsible for morphogenesis, are highly conservative within long evolution periods.

We introduce tropical holonomic sequences of a given order and calculate their entropy in case of the second order.

We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.