# The Newton Polynomial

# The Newton Polynomial

This chapter focuses on the Newton polynomial based on assumption that *K* is a differential-valued field of *H*-type with asymptotic integration and small derivation. Here *K* is also assumed to be equipped with a monomial group and (Γ, ψ) is the asymptotic couple of *K*. Throughout, *P* is an element of *K*{*Y*}superscript Not Equal To. The chapter first revisits the dominant part of *P* before discussing the elementary properties of the Newton polynomial. It then presents results about the shape of the Newton polynomial and considers realizations of three cuts in the value group Γ of *K*. It also describes eventual equalizers, along with further consequences of ω-freeness and λ-freeness, the asymptotic equation over *K*, and some special *H*-fields.

*Keywords:*
differential-valued field, dominant part, Newton polynomial, value group, eventual equalizer, ω-freeness, λ-freeness, asymptotic equation, H-field

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