[Paper Review] A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield

Trevor Gunn|arXiv (Cornell University)|Nov 27, 2019

Polynomial and algebraic computation7 references4 citations

TL;DR

This paper establishes a unified Newton polygon rule for polynomials over formally-real, non-Archimedean, ordered valued fields by introducing a multiplicity formula over the signed tropical hyperfield $\mathbf{TR}$. It proves that the multiplicity of a root with valuation $r$ and positive sign equals the number of sign changes in the initial form $\operatorname{in}_{-r}(p)$, providing an upper bound congruent modulo 2 to the actual number of such roots—generalizing both Descartes’s rule of signs and the classical Newton polygon rule.

ABSTRACT

By defining multiplicities for zeros of polynomials over hyperfields, Baker and Lorscheid were able to provide a unifying perspective on Descartes's rule and the Newton polygon rule for polynomials over a formally-real and valued field respectively. In this paper, we apply their multiplicity formula to the hyperfield associated with formally-real, valued fields to prove a Newton polygon rule which combines Descartes's rule of signs with the classical Newton polygon rule.

Motivation & Objective

• To unify Descartes’s rule of signs and the classical Newton polygon rule for polynomials over formally-real, valued fields.
• To define and compute multiplicities of roots in terms of sign and valuation structure using the signed tropical hyperfield $\mathbf{TR}$.
• To establish a multiplicity formula that bounds the number of real roots with a given sign and valuation, with the error being an even number due to complex conjugate pairs.
• To prove that this multiplicity is maximal and achievable via explicit lifting to a real-closed, non-Archimedean valued field.

Proposed method

• Define multiplicities over the signed tropical hyperfield $\mathbf{TR}$ using a recursive formula analogous to Baker and Lorscheid’s approach over $\mathbf{S}$ and $\mathbf{T}$.
• Prove that the multiplicity $\operatorname{mult}^{\mathbf{TR}}_a(p)$ for $a = (+1, r)$ equals $\Delta(\operatorname{in}_{-r}(p))$, the number of sign changes in the initial form of $p$ at slope $-r$.
• Construct a lifting $P \in \mathbf{R}[[t^{\mathbb{R}}]][x]$ of $p \in \mathbf{TR}[x]$ such that $P$ achieves the maximal number of real roots with specified valuation and sign.
• Use the converse of Descartes’s rule to select lifts $R_\sigma(x)$ of initial forms with exactly the predicted number of positive or negative real roots.
• Ensure the Newton polygon of the lifted polynomial matches that of the original by adjusting the degree via a scaling factor $t^\delta$.
• Verify that the leading terms of roots of $\operatorname{in}_\sigma(P)$ correspond exactly to the roots of $R_\sigma(x)$, thus realizing the multiplicity bound.

Experimental results

Research questions

• RQ1 Can a single multiplicity formula over the signed tropical hyperfield $\mathbf{TR}$ unify Descartes’s rule of signs and the Newton polygon rule?
• RQ2 What is the precise relationship between the number of sign changes in the initial form of a polynomial and the number of real roots with a given valuation and sign?
• RQ3 Is the multiplicity defined over $\mathbf{TR}$ an upper bound on the number of real roots with specified sign and valuation, and what accounts for the difference?
• RQ4 Can a polynomial over $\mathbf{TR}$ be lifted to a real-closed, non-Archimedean valued field such that the maximal number of real roots with given sign and valuation is achieved?

Key findings

• The multiplicity $\operatorname{mult}^{\mathbf{TR}}_a(p)$ for $a = (+1, r)$ equals $\Delta(\operatorname{in}_{-r}(p))$, the number of sign changes in the initial form of $p$ at slope $-r$.
• This multiplicity is an upper bound on the number of roots of any lifting of $p$ to a real-closed, non-Archimedean-ordered valued field with valuation $r$ and positive sign.
• The difference between this upper bound and the actual number of such roots is always an even number, corresponding to complex conjugate pairs.
• For any $p \in \mathbf{TR}[x]$, there exists a lifting $P \in \mathbf{R}[[t^{\mathbb{R}}]][x]$ such that $P$ has exactly $\operatorname{mult}^{\mathbf{TR}}_a(p)$ roots of valuation $|a|$ with positive leading coefficient, and similarly for negative sign via $p(-x)$.
• The construction ensures that the Newton polygon of $P$ matches that of $p$, and the leading terms of the roots of $P$ correspond precisely to the real roots of the initial forms $R_\sigma(x)$.
• The method generalizes Baker and Lorscheid’s results over $\mathbf{S}$ and $\mathbf{T}$ to the signed tropical hyperfield $\mathbf{TR}$, completing the classification of multiplicities over these three key hyperfields.