A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield
arXiv:1911.12274
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.
16 pages, 2 figures, rephrased certain parts of the paper in terms of initial forms for consistency, added an example in the introduction, wrote some motivating exposition at the top of certain sections, definitions are now numbered, subsections 2.{5,6,7} are now a separate section (sec. 3), corrected some minor mistakes