Newton polygon

Summary

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 , over , where 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 of the power series expansion solutions to equations where is a polynomial with coefficients in , the polynomial ring; that is, implicitly defined algebraic functions. The exponents here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in with for a denominator corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating .

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.

DefinitionEdit

 
Construction of the Newton polygon of the polynomial   with respect to the 5-adic valuation.

A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots.

Let   be a field endowed with a non-archimedean valuation  , and let

 

with  . Then the Newton polygon of   is defined to be the lower boundary of the convex hull of the set of points   ignoring the points with  .

Restated geometrically, plot all of these points Pi on the xy-plane. Let's assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P0, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk1 (not necessarily P1). Break the ray here. Now draw a second ray from Pk1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk2. Continue until the process reaches the point Pn; the resulting polygon (containing the points P0, Pk1, Pk2, ..., Pkm, Pn) is the Newton polygon.

Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points.

For a neat diagram of this see Ch6 §3 of "Local Fields" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition.

Main theoremEdit

With the notations in the previous section, the main result concerning the Newton polygon is the following theorem,[1] which states that the valuation of the roots of   are entirely determined by its Newton polygon:

Let   be the slopes of the line segments of the Newton polygon of   (as defined above) arranged in increasing order, and let   be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points   and   then the length is  ).

  • The   are distinct;
  •  ;
  • if   is a root of   in  ,  ;
  • for every  , the number of roots of   whose valuations are equal to   (counting multiplicities) is at most  , with equality if   splits into the product of linear factors over  .

Corollaries and applicationsEdit

With the notation of the previous sections, we denote, in what follows, by   the splitting field of   over  , and by   an extension of   to  .

Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example:

  • Suppose that the valuation   is discrete and normalized, and that the Newton polynomial of   contains only one segment whose slope is   and projection on the x-axis is  . If  , with   coprime to  , then   is irreducible over  . In particular, since the Newton polygon of an Eisenstein polynomial consists of a single segment of slope   connecting   and  , Eisenstein criterion follows.

Indeed, by the main theorem, if   is a root of  ,   If   were not irreducible over  , then the degree   of   would be  , and there would hold  . But this is impossible since   with   coprime to  .

Another simple corollary is the following:

  • Assume that   is Henselian. If the Newton polygon of   fulfills   for some  ,   has a root in  .

Proof: By the main theorem,   must have a single root   whose valuation is   In particular,   is separable over  . If   does not belong to  ,   has a distinct Galois conjugate   over  , with  ,[2] and   is a root of  , a contradiction.

More generally, the following factorization theorem holds:

  • Assume that   is Henselian. Then  , where  ,   is monic for every  , the roots of   are of valuation  , and  .[3]
Moreover,  , and if   is coprime to  ,   is irreducible over  .

Proof: For every  , denote by   the product of the monomials   such that   is a root of   and  . We also denote   the factorization of   in   into prime monic factors   Let   be a root of  . We can assume that   is the minimal polynomial of   over  . If   is a root of  , there exists a K-automorphism   of   that sends   to  , and we have   since   is Henselian. Therefore   is also a root of  . Moreover, every root of   of multiplicity   is clearly a root of   of multiplicity  , since repeated roots share obviously the same valuation. This shows that   divides   Let  . Choose a root   of  . Notice that the roots of   are distinct from the roots of  . Repeat the previous argument with the minimal polynomial of   over  , assumed w.l.g. to be  , to show that   divides  . Continuing this process until all the roots of   are exhausted, one eventually arrives to  , with  . This shows that  ,   monic. But the   are coprime since their roots have distinct valuations. Hence clearly  , showing the main contention. The fact that   follows from the main theorem, and so does the fact that  , by remarking that the Newton polygon of   can have only one segment joining   to  . The condition for the irreducibility of   follows from the corollary above. (q.e.d.)

The following is an immediate corollary of the factorization above, and constitutes a test for the reducibility of polynomials over Henselian fields:

  • Assume that   is Henselian. If the Newton polygon does not reduce to a single segment   then   is reducible over  .

Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a Newton polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like  

 
This diagram shows the Newton polygon for P(x,y) = 3x2 y3xy2 + 2x2y2x3y, with positive monomials in red and negative monomials in cyan. Faces are labelled with the limiting terms they correspond to.

Symmetric function explanationEdit

In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual roots, in an algebraic closure. This has aspects both of ramification theory and singularity theory. The valid inferences possible are to the valuations of power sums, by means of Newton's identities.

HistoryEdit

Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg.[4]

See alsoEdit

ReferencesEdit

  1. ^ For an interesting demonstration based on hyperfields, see Matthew Baker, Oliver Lorscheid, (2021). Descartes' rule of signs, Newton polygons, and polynomials over hyperfields.Journal of Algebra, Volume 569, p. 416-441.
  2. ^ Recall that in Henselian rings, any valuation extends uniquely to every algebraic extension of the base field. Hence   extends uniquely to  . But   is an extension of   for every automorphism   of  , therefore  
  3. ^ J. W. S. Cassels, Local Fields, Chap. 6, thm. 3.1.
  4. ^ Egbert Brieskorn, Horst Knörrer (1986). Plane Algebraic Curves, pp. 370–383.
  • Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61480-4, ISBN 978-3-540-61087-8, MR 1423131
  • Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199.

External linksEdit

  • Applet drawing a Newton Polygon