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.
With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, which states that the valuation of the roots of are entirely determined by its Newton polygon:
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 , 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 .
Moreover, , and if is coprime to , is irreducible over .
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) = 3x2y3 − xy2 + 2x2y2 − x3y, with positive monomials in red and negative monomials in cyan. Faces are labelled with the limiting terms they correspond to.
^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.
^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
^ J. W. S. Cassels, Local Fields, Chap. 6, thm. 3.1.
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.
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