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Netanel Friedenberg

Publications and source records attributed to Netanel Friedenberg.

10 recordsLinked to original sources
arXiv2026

Integral closure for (additively idempotent) semirings

In commutative ring theory there are multiple equivalent definitions of integrality. These notions diverge when working with idempotent semirings. In this paper we present these different definitions of integrality for semirings and explore the relations between them. As a tool, we prove a Cayley-Hamilton theorem over additively idempotent semirings, which may be of broader interest. In examples, we compute integral closures of coordinate semirings in their total semiring of fractions and integral closures of sub-semirings of coordinate semirings. Such computation gives avenues to defining and understanding the normalization of tropical varieties as well as computing normalization of varieties tropically.

math.AC↗
arXiv2026

Geometric classification of primes modulo a (bend) congruence

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 of this result. (1) We prove an analogue of the strong Nullstellensatz for congruences with finite tropical basis. This extends the existing result of Jo\'o-Mincheva to cases, such as the bend congruence of a tropical(ized) ideal, where the congruence is not finitely generated. (2) We show that, if $I$ is the ideal of an affine variety not contained in the coordinate hyperplanes, then $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is cancellative. This result has applications to the integral closure (as per Tolliver) of $\mathbb{T}[x_1, \dots, x_n]/\operatorname{Bend}(\operatorname{trop} I)$ which we explore in a forthcoming paper. (3) We show that $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is the tropical function semiring on $\operatorname{trop} V(I)$, which creates a bridge between the algebraic approach to non-embedded tropicalization in the work of J. Song and the bend congruence approach of Giansiracusa-Giansiracusa and Maclagan-Rinc\'on. (4) As a consequence of one of our lemmas, we describe the closure of a polyhedron in a tropical toric variety even when the polyhedron is not compatible with the fan defining the tropical toric variety.

math.AG↗
arXiv2023

Geometric interpretation of valuated term (pre)orders

Valuated term orders are studied for the purposes of Gr\"{o}bner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry.

math.AG↗
arXiv2023

A construction of algebraizable formal models

Let $X$ be a variety over a complete nontrivially valued field $K$. We construct an algebraizable formal model for the analytification of $X$ in the case $X$ admits a closed embedding into a toric variety. By algebraizable we mean that the formal model is given by the completion along the special fiber of a locally finite type flat scheme over the valuation ring $K^\circ$. We construct the formal model via the combinatorial theory of $\mathbb{T}$-toric varieties over $K^{\circ}$.

math.AG↗
arXiv2023

Locally finite completions of polyhedral complexes

We develop a method for subdividing polyhedral complexes in a way that restricts the possible recession cones and allows one to work with a fixed class of polyhedron. We use these results to construct locally finite completions of rational polyhedral complexes whose recession cones lie in a fixed fan, locally finite polytopal completions of polytopal complexes, and locally finite zonotopal completions of zonotopal complexes.

math.CO↗
arXiv2022

Tropical adic spaces I: The continuous spectrum of a topological semiring

Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow up paper.

math.AG↗
arXiv2019

Normal completions of toric varieties over rank one valuation rings and completions of $\Gamma$-admissible fans

We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma$-admissible fans admit $\Gamma$-admissible completions. In order to show this we prove a combinatorial analog of noetherian reduction which we believe will be of independent interest.

math.AG↗
arXiv2017

Two-vertex generators of Jacobians of graphs

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.

math.CO↗