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Source: 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 ultrametric 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

K
[
[
X
]
]

{\displaystyle K[[X]]}

,
over

K

{\displaystyle K}

, where

K

{\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

a

X

r

{\displaystyle aX^{r}}

of the power series expansion solutions to equations

P
(
F
(
X
)
)
=
0

{\displaystyle P(F(X))=0}

where

P

{\displaystyle P}

is a polynomial with coefficients in

K
[
X
]

{\displaystyle K[X]}

, the polynomial ring; that is, implicitly defined algebraic functions. The exponents

r

{\displaystyle r}

here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in

K
[
[
Y
]
]

{\displaystyle K[[Y]]}

with

Y
=

X

1
d

{\displaystyle Y=X^{\frac {1}{d}}}

for a denominator

d

{\displaystyle d}

corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating

d

{\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.

Definition

Newton

Main theorem

Corollaries and applications

, and there would hold

v

L

(
α
)
∈

1
d

Z

{\displaystyle v_{L}(\alpha )\in {1 \over d}\mathbb {Z} }

. But this is impossible since

v

L

(
α
)
=
−
a

/

n

{\displaystyle v_{L}(\alpha )=-a/n}

with

a

{\displaystyle a}

coprime to

n

{\displaystyle n}

.
Another simple corollary is the following:

Assume that

(
K
,

v

K

)

{\displaystyle (K,v_{K})}

is Henselian. If the Newton polygon of

f

{\displaystyle f}

fulfills

λ

i

=
1

{\displaystyle \lambda _{i}=1}

for some

i

{\displaystyle i}

, then

f

{\displaystyle f}

has a root in

K

{\displaystyle K}

.
Proof: By the main theorem,

f

{\displaystyle f}

must have a single root

α

{\displaystyle \alpha }

whose valuation is

v

L

(
α
)
=
−

μ

i

.

{\displaystyle v_{L}(\alpha )=-\mu _{i}.}

In particular,

α

{\displaystyle \alpha }

is separable over

K

{\displaystyle K}

.
If

α

{\displaystyle \alpha }

does not belong to

K

{\displaystyle K}

,

α

{\displaystyle \alpha }

has a distinct Galois conjugate

α
′

{\displaystyle \alpha '}

over

K

{\displaystyle K}

, with

v

L

(

α
′

)
=

v

L

(
α
)

{\displaystyle v_{L}(\alpha ')=v_{L}(\alpha )}

, and

α
′

{\displaystyle \alpha '}

is a root of

f

{\displaystyle f}

, a contradiction.
More generally, the following factorization theorem holds:

Assume that

(
K
,

v

K

)

{\displaystyle (K,v_{K})}

is Henselian. Then

f
=
A

f

1

f

2

⋯

f

r

,

{\displaystyle f=A\,f_{1}\,f_{2}\cdots f_{r},}

, where

A
∈
K

{\displaystyle A\in K}

,

f

i

∈
K
[
X
]

{\displaystyle f_{i}\in K[X]}

is monic for every

i

{\displaystyle i}

, the roots of

f

i

{\displaystyle f_{i}}

are of valuation

−

μ

i

{\displaystyle -\mu _{i}}

, and

deg
⁡
(

f

i

)
=

λ

i

{\displaystyle \deg(f_{i})=\lambda _{i}}

.
Moreover,

μ

i

=

v

K

(

f

i

(
0
)
)

/

λ

i

{\displaystyle \mu _{i}=v_{K}(f_{i}(0))/\lambda _{i}}

, and if

v

K

(

f

i

(
0
)
)

{\displaystyle v_{K}(f_{i}(0))}

is coprime to

λ

i

{\displaystyle \lambda _{i}}

,

f

i

{\displaystyle f_{i}}

is irreducible over

K

{\displaystyle K}

.
Proof:
For every

i

{\displaystyle i}

, denote by

f

i

{\displaystyle f_{i}}

the product of the monomials

(
X
−
α
)

{\displaystyle (X-\alpha )}

such that

α

{\displaystyle \alpha }

is a root of

f

{\displaystyle f}

and

v

L

(
α
)
=
−

μ

i

{\displaystyle v_{L}(\alpha )=-\mu _{i}}

. We also denote

f
=
A

P

1

k

1

P

2

k

2

⋯

P

s

k

s

{\displaystyle f=AP_{1}^{k_{1}}P_{2}^{k_{2}}\cdots P_{s}^{k_{s}}}

the factorization of

f

{\displaystyle f}

in

K
[
X
]

{\displaystyle K[X]}

into prime monic factors

(
A
∈
K
)
.

{\displaystyle (A\in K).}

Let

α

{\displaystyle \alpha }

be a root of

f

i

{\displaystyle f_{i}}

. We can assume that

P

1

{\displaystyle P_{1}}

is the minimal polynomial of

α

{\displaystyle \alpha }

over

K

{\displaystyle K}

.
If

α
′

{\displaystyle \alpha '}

is a root of

P

1

{\displaystyle P_{1}}

, there exists a K-automorphism

σ

{\displaystyle \sigma }

of

L

{\displaystyle L}

that sends

α

{\displaystyle \alpha }

to

α
′

{\displaystyle \alpha '}

, and we have

v

L

(
σ
α
)
=

v

L

(
α
)

{\displaystyle v_{L}(\sigma \alpha )=v_{L}(\alpha )}

since

K

{\displaystyle K}

is Henselian. Therefore

α
′

{\displaystyle \alpha '}

is also a root of

f

i

{\displaystyle f_{i}}

.
Moreover, every root of

P

1

{\displaystyle P_{1}}

of multiplicity

ν

{\displaystyle \nu }

is clearly a root of

f

i

{\displaystyle f_{i}}

of multiplicity

k

1

ν

{\displaystyle k_{1}\nu }

, since repeated roots share obviously the same valuation. This shows that

P

1

k

1

{\displaystyle P_{1}^{k_{1}}}

divides

f

i

.

{\displaystyle f_{i}.}

Let

g

i

=

f

i

/

P

1

k

1

{\displaystyle g_{i}=f_{i}/P_{1}^{k_{1}}}

. Choose a root

β

{\displaystyle \beta }

of

g

i

{\displaystyle g_{i}}

. Notice that the roots of

g

i

{\displaystyle g_{i}}

are distinct from the roots of

P

1

{\displaystyle P_{1}}

. Repeat the previous argument with the minimal polynomial of

β

{\displaystyle \beta }

over

K

{\displaystyle K}

, assumed w.l.g. to be

P

2

{\displaystyle P_{2}}

, to show that

P

2

k

2

{\displaystyle P_{2}^{k_{2}}}

divides

g

i

{\displaystyle g_{i}}

.
Continuing this process until all the roots of

f

i

{\displaystyle f_{i}}

are exhausted, one eventually arrives to

f

i

=

P

1

k

1

⋯

P

m

k

m

{\displaystyle f_{i}=P_{1}^{k_{1}}\cdots P_{m}^{k_{m}}}

, with

m
≤
s

{\displaystyle m\leq s}

. This shows that

f

i

∈
K
[
X
]

{\displaystyle f_{i}\in K[X]}

,

f

i

{\displaystyle f_{i}}

monic.
But the

f

i

{\displaystyle f_{i}}

are coprime since their roots have distinct valuations. Hence clearly

f
=
A

f

1

⋅

f

2

⋯

f

r

{\displaystyle f=Af_{1}\cdot f_{2}\cdots f_{r}}

, showing the main contention.
The fact that

λ

i

=
deg
⁡
(

f

i

)

{\displaystyle \lambda _{i}=\deg(f_{i})}

follows from the main theorem, and so does the fact that

μ

i

=

v

K

(

f

i

(
0
)
)

/

λ

i

{\displaystyle \mu _{i}=v_{K}(f_{i}(0))/\lambda _{i}}

, by remarking that the Newton polygon of

f

i

{\displaystyle f_{i}}

can have only one segment joining

(
0
,

v

K

(

f

i

(
0
)
)

{\displaystyle (0,v_{K}(f_{i}(0))}

to

(

λ

i

,
0
=

v

K

(
1
)
)

{\displaystyle (\lambda _{i},0=v_{K}(1))}

. The condition for the irreducibility of

f

i

{\displaystyle 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

)

{\displaystyle (K,v_{K})}

is Henselian. If the Newton polygon does not reduce to a single segment

(
μ
,
λ
)
,

{\displaystyle (\mu ,\lambda ),}

then

f

{\displaystyle f}

is reducible over

K

{\displaystyle K}

.
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.

{\displaystyle 3x^{2}y^{3}-xy^{2}+2x^{2}y^{2}-x^{3}y=0.}

Symmetric function explanation

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.

History

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.

References

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 links

Applet drawing a Newton Polygon

Definition

Main theorem

Corollaries and applications

Symmetric function explanation

History

See also

References

External links

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