Newton polygon

In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring ${\displaystyle K[[X]]}$, over ${\displaystyle K}$, where ${\displaystyle K}$ was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms ${\displaystyle aX^{r}}$ of the power series expansion solutions to equations ${\displaystyle P(F(X))=0}$ where ${\displaystyle P}$ is a polynomial with coefficients in ${\displaystyle K[X]}$, the polynomial ring; that is, implicitly defined algebraic functions. The exponents ${\displaystyle r}$ here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in ${\displaystyle K[[Y]]}$ with ${\displaystyle Y=X^{\frac {1}{d}}}$ for a denominator ${\displaystyle d}$ corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating ${\displaystyle d}$.

After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.