= Newton polygon root valuation theorem
{c}
{title2=$\text{slope}=-v(\text{root})$}
= Newton polygon theorem
{c}
{synonym}
For a <polynomial> with nonzero constant coefficient over a complete <discretely valued field>, plot coefficient exponents against their <valuations>. A lower <Newton polygon> segment of slope $s$ and horizontal length $r$ accounts for exactly $r$ roots of <valuation> $-s$, counted with multiplicity in an <algebraic closure>. Zero coefficients are omitted. Reflecting the horizontal axis changes the sign of every slope, so slope conventions must be fixed before reading root <valuations>.
Back to article page
