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

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