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Algebraic Solution of Tropical Polynomial Optimization Problems
Mathematics, 2021 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).
Nikolai Krivulin
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Algebraic Solution of Tropical Best Approximation Problems
Mathematics, 2023 We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition.
Nikolai Krivulin
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Tropical Newton–Puiseux Polynomials [PDF]
Lecture Notes in Computer Science, 2018 We introduce tropical Newton-Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton-Puiseux polynomial. A polynomial complexity algorithm for resolubility of a tropical curve is designed.
Dima Grigoriev, Grigoriev Dima
exaly +5 more sources
Tropical Newton-Puiseux polynomials II
Journal of Symbolic Computation, 2023 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.
exaly +4 more sources
Convergent Hahn series and tropical geometry of higher rank
Journal of the London Mathematical Society, Volume 107, Issue 4, Page 1450-1481, April 2023., 2023 Abstract We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one tropical polytopes, which is beneficial for algorithmic purposes.
Michael Joswig, Ben Smith
wiley +1 more source
Bulletin of the London Mathematical Society, Volume 54, Issue 4, Page 1249-1264, August 2022., 2022 Abstract We define a tropical version Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ of the Fano Scheme Fd(X)$\operatorname{F}_d(X)$ of a projective variety X⊆Pn$X\subseteq \mathbb {P}^n$ and prove that Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ is the support of a polyhedral complex contained in tropG(d,n)$\operatorname{trop}\mathbb {G ...
Sara Lamboglia
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Tropical Solution of Discrete Best Approximation Problems
MathematicsWe consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input–output pairs of an unknown function defined on a
Nikolai Krivulin
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A tropical approach to rigidity: Counting realisations of frameworks
Journal of the London Mathematical Society, Volume 113, Issue 2, February 2026.Abstract A realisation of a graph in the plane as a bar‐joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points.
Oliver Clarke +6 more
wiley +1 more source
Moments, sums of squares, and tropicalization
Journal of the London Mathematical Society, Volume 112, Issue 4, October 2025.Abstract We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set S$S$. The truncated cones of moments of measures supported on the set S$S$ are dual to nonnegative polynomials on S$S$, while “pseudomoments” are dual to sums of squares approximations to nonnegative polynomials.
Grigoriy Blekherman +4 more
wiley +1 more source
Generic root counts and flatness in tropical geometry
Journal of the London Mathematical Society, Volume 111, Issue 5, May 2025.Abstract We use tropical and nonarchimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space Y$Y$. In particular, we are interested in the choices of parameters for which the generic root count is attained.
Paul Alexander Helminck, Yue Ren
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