Newton polygon

Image:Newton-polygon.gif

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 $K$ be a field endowed with a non-archimedean valuation $v\_K:\; K\; \backslash to\; \backslash mathbb\; R\backslash cup\; \backslash \{\; \backslash infty\; \backslash \}$, and let

:$f(x)\; =\; a\_nx^n\; +\; \backslash cdots\; +\; a\_1x\; +\; a\_0\; \backslash in\; K[x],$

with $a\_0\; a\_n\; \backslash ne\; 0$. Then the Newton polygon of $f$ is defined to be the lower boundary of the convex hull of the set of points $P\_i=\backslash left(i,v\_K(a\_i)\backslash right),$

ignoring the points with $a\_i\; =\; 0$.

Restated geometrically, plot all of these points P_i on the xy-plane. Let's assume that the points indices increase from left to right (P₀ is the leftmost point, P_n 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 P_k1 (not necessarily P₁). Break the ray here. Now draw a second ray from P_k1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P_k2. Continue until the process reaches the point P_n; the resulting polygon (containing the points P₀, P_k1, P_k2, ..., P_km, P_n) is the Newton polygon.

Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P₀, ..., P_n. 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,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. which states that the valuation of the roots of $f$ are entirely determined by its Newton polygon:

Let $\backslash mu\_1,\; \backslash mu\_2,\; \backslash ldots,\; \backslash mu\_r$

be the slopes of the line segments of the Newton polygon of $f(x)$ (as defined above) arranged in increasing order, and let

$\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_r$

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 $P\_i$ and $P\_j$ then the length is $j-i$).

• The $\backslash mu\_i$ are distinct;
• $\backslash sum\_i\; \backslash lambda\_i\; =\; n$;
• if $\backslash alpha$ is a root of $f$ in $K$, $v(\backslash alpha)\; \backslash in\; \backslash \{-\backslash mu\_1,\; \backslash ldots\; ,\; -\backslash mu\_r\backslash \}$;
• for every $i$, the number of roots of $f$ whose valuations are equal to $-\backslash mu\_i$ (counting multiplicities) is at most $\backslash lambda\_i$, with equality if $f$ splits into the product of linear factors over $K$.

With the notation of the previous sections, we denote, in what follows, by $L$ the splitting field of $f$ over $K$, and by $v\_L$ an extension of $v\_K$ to $L$.

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

• Suppose that the valuation $v$ is discrete and normalized, and that the Newton polynomial of $f$ contains only one segment whose slope is $\backslash mu$ and projection on the x-axis is $\backslash lambda$. If $\backslash mu\; =\; a/n$, with $a$ coprime to $n$, then $f$ is irreducible over $K$. In particular, since the Newton polygon of an Eisenstein polynomial consists of a single segment of slope $-\backslash frac\{1\}\{n\}$ connecting $(0,1)$ and $(n,0)$, Eisenstein criterion follows.

Indeed, by the main theorem, if $\backslash alpha$ is a root of $f$, $v\_L(\backslash alpha)\; =\; -a/n.$

If $f$ were not irreducible over $K$, then the degree $d$ of $\backslash alpha$ would be , and there would hold $v\_L(\backslash alpha)\; \backslash in\; \{1\backslash over\; d\}\backslash mathbb\; Z$. But this is impossible since $v\_L(\backslash alpha)\; =\; -a/n$ with $a$ coprime {{nobr|to $n$}}.

Another simple corollary is the following:

• Assume that $(K,\; v\_K)$ is Henselian. If the Newton polygon of $f$ fulfills $\backslash lambda\_i\; =\; 1$ for some $i$, then $f$ has a root in $K$.

Proof: By the main theorem, $f$ must have a single root $\backslash alpha$ whose valuation is $v\_L(\backslash alpha)\; =\; -\backslash mu\_i.$ In particular, $\backslash alpha$ is separable over $K$.

If $\backslash alpha$ does not belong to $K$, $\backslash alpha$ has a distinct Galois conjugate $\backslash alpha\text{'}$ over $K$, with $v\_L(\backslash alpha\text{'})\; =\; v\_L(\backslash alpha)$,Recall that in Henselian rings, any valuation extends uniquely to every algebraic extension of the base field. Hence $v\_K$ extends uniquely to $v\_L$. But $v\_L\backslash circ\; \backslash sigma$ is an extension of $v\_K$ for every automorphism $\backslash sigma$ of $L$, therefore $v\_L(\backslash alpha\text{'})\; =\; v\_L\backslash circ\; \backslash sigma\; (\backslash alpha)\; =\; v\_L(\backslash alpha).$ and $\backslash alpha\text{'}$ is a root of $f$, a contradiction.

More generally, the following factorization theorem holds:

• Assume that $(K,\; v\_K)$ is Henselian. Then $f\; =\; A\backslash ,f\_1\backslash ,\; f\_2\backslash cdots\; f\_r,$, where $A\backslash in\; K$, $f\_i\backslash in\; K[X]$ is monic for every $i$, the roots of $f\_i$ are of valuation $-\backslash mu\_i$, and $\backslash deg(f\_i)\; =\; \backslash lambda\_i$. J. W. S. Cassels, Local Fields, Chap. 6, thm. 3.1.

:Moreover, $\backslash mu\_i\; =\; v\_K(f\_i(0))/\backslash lambda\_i$, and if $v\_K(f\_i(0))$ is coprime to $\backslash lambda\_i$, $f\_i$ is irreducible over $K$.

Proof:

For every $i$, denote by $f\_i$ the product of the monomials $(X\; -\; \backslash alpha)$ such that $\backslash alpha$ is a root of $f$ and $v\_L(\backslash alpha)\; =\; -\backslash mu\_i$. We also denote $f\; =\; A\; P\_1^\{k\_1\}P\_2^\{k\_2\}\backslash cdots\; P\_s^\{k\_s\}$ the factorization of $f$ in $K[X]$ into prime monic factors $(A\backslash in\; K).$

Let $\backslash alpha$ be a root of $f\_i$. We can assume that $P\_1$ is the minimal polynomial of $\backslash alpha$ over $K$.

If $\backslash alpha\text{'}$ is a root of $P\_1$, there exists a K-automorphism $\backslash sigma$ of $L$ that sends $\backslash alpha$ to $\backslash alpha\text{'}$, and we have $v\_L(\backslash sigma\; \backslash alpha)\; =\; v\_L(\backslash alpha)$ since $K$ is Henselian. Therefore $\backslash alpha\text{'}$ is also a root of $f\_i$.

Moreover, every root of $P\_1$ of multiplicity $\backslash nu$ is clearly a root of $f\_i$ of multiplicity $k\_1\backslash nu$, since repeated roots share obviously the same valuation. This shows that $P\_1^\{k\_1\}$ divides $f\_i.$

Let $g\_i\; =\; f\_i/P\_1^\{k\_1\}$. Choose a root $\backslash beta$ of $g\_i$. Notice that the roots of $g\_i$ are distinct from the roots of $P\_1$. Repeat the previous argument with the minimal polynomial of $\backslash beta$ over $K$, assumed w.l.g. to be $P\_2$, to show that $P\_2^\{k\_2\}$ divides $g\_i$.

Continuing this process until all the roots of $f\_i$ are exhausted, one eventually arrives to

$f\_i\; =\; P\_1^\{k\_1\}\backslash cdots\; P\_m^\{k\_m\}$, with $m\; \backslash leq\; s$. This shows that $f\_i\backslash in\; K[X]$, $f\_i$ monic.

But the $f\_i$ are coprime since their roots have distinct valuations. Hence clearly $f\; =\; A\; f\_1\backslash cdot\; f\_2\backslash cdots\; f\_r$, showing the main contention.

The fact that $\backslash lambda\_i\; =\; \backslash deg(f\_i)$ follows from the main theorem, and so does the fact that $\backslash mu\_i\; =\; v\_K(f\_i(0))/\backslash lambda\_i$, by remarking that the Newton polygon of $f\_i$ can have only one segment joining $(0,\; v\_K(f\_i(0))$ to $(\backslash lambda\_i,\; 0\; =\; v\_K(1))$. The condition for the irreducibility of $f\_i$ 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 $(K,\; v\_K)$ is Henselian. If the Newton polygon does not reduce to a single segment $(\backslash mu,\; \backslash lambda),$ then $f$ is reducible over $K$.

[[Image:Diagram_of_a_Newton_Polygon_Convex_hull.svg|thumb|right| The Newton polygon for {{nowrap|1= 3x² y³ − xy² + 2x²y² − x³y}} with positive monomials in red and negative monomials in cyan. Faces are labelled with their limiting terms.

]]

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

$3\; x^2\; y^3\; -\; x\; y^2\; +\; 2\; x^2\; y^2\; -\; x^3\; y\; =\; 0.$

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.

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.Egbert Brieskorn, Horst Knörrer (1986). Plane Algebraic Curves, pp. 370–383.

• F-crystal
• Eisenstein's criterion
• Newton–Okounkov body
• Newton polytope

{{reflist}}

• {{Citation | last=Goss | first=David | title=Basic structures of function field arithmetic | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-61087-8 |mr=1423131 | year=1996 | volume=35 | doi=10.1007/978-3-642-61480-4}}
• Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199.

{{Commons category|Newton polygon}}

• [http://www.math.sc.edu/~filaseta/newton/newton.html Applet drawing a Newton Polygon]

{{Isaac Newton}}

Category:Algebraic number theory

Category:Symmetric functions

Category:Isaac Newton