Tropical Polynomial Research Articles

Asymmetric Cryptography Based on the Tropical Jones Matrix

In recent years, the tropical polynomial factorization problem, the tropical matrix decomposition problem, and the tropical multivariate quadratic equation solving problem have been proved to be NP-hard. Some asymmetric cryptographic systems based on tropical semirings have been proposed, but most of them are insecure and have been successfully attacked. In this paper, a new key exchange protocol and a new encryption protocol are proposed based on the difficulty of finding the multiple exponentiation problem of the tropical Jones matrices. The analysis results indicate that our protocol can resist various existing attacks. The complexity of attacking an MEP by adversaries is raised due to the larger number of combinations in the tropical Jones matrices compared to regular matrix polynomials. Furthermore, the index semiring is the non-negative integer cyclic matrix semiring, leading to a higher efficiency in key generation.

Some statistics about Tropical Sandpile Model

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number $n$ of randomly dropped points $P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropical polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap \partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points $P$ removed, is a tree.

Tropical algebra for noise removal and optimal control

Algorithms for noise removal are either complex or ineffective, and the optimal control with inequality constrains makes the algorithm even more complex. Now the condition is changed completely, and the tropical algebra is an extremely simple tool for this purpose. The tropical algebra-based filter has obvious advantages over traditional ones, and an inequality constrain can be converted to a tropical polynomial, making the tropical algebra much attractive in engineering applications. In this paper, idempotents on multiplicative semigroups of tropical matrices are studied. First, the concepts of tropical algebra and the semigroup of tropical matrices under multiplication are introduced. Second, the structure of idempotents on the semigroup of [Formula: see text] tropical matrices under multiplication is given. Finally, as examples, the tropical addition is used as a filter for noise removal, and a simple optimization is given where the constraint is given by the tropical algebra. The paper has opened the path for a new way to noise removal and optimization.

Tropical Newton-Puiseux polynomials II

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

Paving tropical ideals

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size \(\deg (I)\) or \(\deg (I)+1\)—we call them paving tropical ideals. We show that paving tropical ideals of degree \(d+1\) are in bijection with \({\mathbb {Z}}^n\)-invariant d-partitions of \({\mathbb {Z}}^n\). This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with \({\mathbb {Z}}^n\)-invariant 2-partitions of quotient groups of the form \({\mathbb {Z}}^n/L\). We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals.

Algebraic Solution of Tropical Polynomial Optimization Problems

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 problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux polynomials, subject to box constraints on the variables. A technique for variable elimination is presented that converts the original optimization problem to a new one in which one variable is removed and the box constraint for this variable is modified. The novel approach may be thought of as an extension of the Fourier–Motzkin elimination method for systems of linear inequalities in ordered fields to the issue of polynomial optimization in ordered tropical semifields. We use this technique to develop a procedure to solve the problem in a finite number of iterations. The procedure includes two phases: backward elimination and forward substitution of variables. We describe the main steps of the procedure, discuss its computational complexity and present numerical examples.

On a tropical version of the Jacobian conjecture

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.

A characterization for tropical polynomials being the minimum finishing time of project networks

A tropical polynomial is called $R$-polynomial if it can be realized as the minimum finishing time of a project network. $R$-polynomials satisfy the term extendability condition, and correspond to simple graphs. We give a characterization of $R$-polynomials in terms of simple graphs.

Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of “tropical roots” depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for “fewnomials” or when the tropical roots are well separated.

Symmetric and r-symmetric tropical polynomials and rational functions

A tropical polynomial in nr variables, divided into blocks of r variables each, is r-symmetric if it is invariant under the action of Sn that permutes the blocks. For r=1 we call these symmetric tropical polynomials. We can define r-symmetric and symmetric tropical rational functions in a similar manner. In this paper we identify generators for the sets of symmetric tropical polynomials and rational functions. While r-symmetric tropical polynomials are not finitely generated for r≥2, we show that r-symmetric tropical rational functions are and provide a list of generators.

Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

Given convex polytopes $$P_1,\ldots ,P_r \subset \mathbb {R}^n$$ P 1 , … , P r ⊂ R n and finite subsets $$\mathcal {W}_I$$ W I of the Minkowski sums $$P_I=\sum _{i \in I} P_i$$ P I = ∑ i ∈ I P i , we consider the quantity $$N(\mathbf {W})=\sum _{I \subset \mathbf{[}r \mathbf{]} } {(-1)}^{r-|I|} \big | \mathcal {W}_I \big |$$ N ( W ) = ∑ I ⊂ [ r ] ( - 1 ) r - | I | | W I | . If $$\mathcal {W}_I=\mathbb {Z}^n \cap P_I$$ W I = Z n ∩ P I and $$P_1,\ldots ,P_n$$ P 1 , … , P n are lattice polytopes in $$\mathbb {R}^n$$ R n , then $$N(\mathbf {W})$$ N ( W ) is the classical mixed volume of $$P_1,\ldots ,P_n$$ P 1 , … , P n giving the number of complex solutions of a general complex polynomial system with Newton polytopes $$P_1,\ldots ,P_n$$ P 1 , … , P n . We develop a technique that we call irrational mixed decomposition which allows us to estimate $$N(\mathbf {W})$$ N ( W ) under some assumptions on the family $$\mathbf {W}=(\mathcal {W}_I)$$ W = ( W I ) . In particular, we are able to show the nonnegativity of $$N(\mathbf {W})$$ N ( W ) in some important cases. A special attention is paid to the family $$\mathbf {W}=(\mathcal {W}_I)$$ W = ( W I ) defined by $$\mathcal {W}_I=\sum _{i \in I} \mathcal {W}_i$$ W I = ∑ i ∈ I W i , where $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ W 1 , … , W r are finite subsets of $$P_1,\ldots ,P_r$$ P 1 , … , P r . The associated quantity $$N(\mathbf {W})$$ N ( W ) is called discrete mixed volume of $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ W 1 , … , W r . Using our irrational mixed decomposition technique, we show that for $$r=n$$ r = n the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ W 1 , … , W n ⊂ R n . We also prove that the discrete mixed volume associated with $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ W 1 , … , W r is bounded from above by the Kouchnirenko number $$\prod _{i=1}^r (|\mathcal {W}_i|-1)$$ ∏ i = 1 r ( | W i | - 1 ) . For $$r=n$$ r = n this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ W 1 , … , W n ⊂ R n . This conjecture was disproved, but our result show that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.

Complexity of tropical Schur polynomials

We study the complexity of computation of a tropical Schur polynomial Tsλ where λ is a partition, and of a tropical polynomial Tmλ obtained by the tropicalization of the monomial symmetric function mλ. Then Tsλ and Tmλ coincide as tropical functions (so, as convex piece-wise linear functions), while differ as tropical polynomials. We prove the following bounds on the complexity of computing over the tropical semi-ring (R,max⁡,+):•a polynomial upper bound for Tsλ and•an exponential lower bound for Tmλ. Also the complexity of tropical skew Schur polynomials is discussed.

On a tropical dual Nullstellensatz

Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical dual Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials.

Detecting integral polyhedral functions

We study the class of real-valued functions on convex subsets of ℝn which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xiao). We also prove that a function on ℝ2 is a tropical polynomial if and only if its restriction to each translate of a generic tropical line is a tropical polynomial.