Ramification of polynomials

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Master’s thesis

Ramification of polynomials

Author: Ana Strikić

Supervisor: Jonas Nordqvist Examiner: Karl-Olof Lindahl Date: 2021-05-26

Course Code: 5MA41E Subject: Mathematics Level: Advanced level Department Of Mathematics

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iii

LINNAEUS UNIVERSITY

Abstract

Faculty of Technology Department of Mathematics

Master of Science

Ramification of polynomials by Ana Strikic

In this research, we study iterations of non-pleasantly ramified polynomials over fields of positive characteristic and subsequently, their lower ramification numbers.

Of particular interest for this thesis are polynomials for which both the multiplicity and the degree of its iterates grow exponentially. Specifically we consider the family of polynomials such that given a positive integer k the family is given by P(z) = z(1+z^3k²⁻¹ +z³^k⁻¹). The cubic polynomial z+z²+z³ is a special case of this family and is particularly interesting.

Keywords: non-pleasantly ramified polynomials, minimally ramified polynomial, lower ramification numbers, minimal ramification, discrete dynamical systems, iterating polynomials, ultrametric fields

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v

1. Introduction

In this thesis we will study iterations of polynomials defined over fields of positive characteristic. In particular, we will study polynomials which have exponential growth on its so-called lower ramification numbers. As such, we are interested in discrete dynamical systems.

A discrete dynamical system on a set X is a pair (X, f), where f is a function f : X → X. Let f⁰ := Id_X then for an integer n ≥1 we define the n-th iterate of f to be fⁿ= fⁿ⁻¹◦ f . Then for a point z0∈ X we can define the forward orbit of z₀ to be the set

{fⁿ(z₀): n≥0}.

Hence, studying the iterations of a polynomial f is closely related to studying the forward orbits of elementsz0.

Suppose that ϕ(z) = αz+β is an affine map, and that g(z) = ϕ⁻¹◦ f◦ϕ(z) is a linear change of coordinates of f . Then we can establish that

gⁿ(z) = (ϕ⁻¹◦f◦ϕ(z)) ◦ (ϕ⁻¹◦f ◦ϕ(z)) ◦ · · · ◦ (ϕ⁻¹◦ f◦ϕ(z)), which simplifies to

gⁿ(z) =ϕ⁻¹◦fⁿ◦ϕ(z).

The dynamical systems f and g are, as such, regarded as equivalent, given that they differ by a change of coordinates ϕ. Hence, the forward orbits of f can be described in terms of g and vice versa.

As mentioned, we will consider the case that f is a polynomial. In this thesis we are interested in adding an algebraic structure to X, in particular, a field of positive characteristic, i.e. algebraic dynamics (see [Sil07; KA09; Ben19]). More specifically, iterated polynomials over fields of positive characteristic are studied in, for example, [Pez94]. This topic is of particular interest in the field of algebraic dynamics as recent results show that there exists a connection between the geo- metric location of periodic points of power series over ultrametric fields and their lower ramification numbers (see [LRL16a; LRL16b; Lin13]).

We are interested in a property called the multiplicity of fixed points and its behaviour under iterations. Let f be a polynomial, then any point z₀ satisfying f(z₀) = z₀ is said to be a fixed point of f . The multiplicity of z₀ as a fixed point can be defined as follows.

Definition 1. Let f(z) be a polynomial with coefficients in field K and z0 be a point such that f(z₀) =z₀. By the Factor theorem it is possible to write f(z) as

f(z) −z= a(z−z0)^q⁰f0(z),

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2 Chapter 1. Introduction

where f₀(z₀) 6= z₀. The unique positive integer q₀ is called the multiplicity of z₀ and is denoted as mult(f , z₀). For z₀=0 the notation is simplified to mult(f).

It is known that in characteristic 0 the multiplicity of polynomial f is fixed under iterations (see Lemma 5.1.1). However, in positive characteristic this statement is false, and whenever the iterate is divisible by the field characteristic the multiplicity of the iterate will be larger than that of f .

In positive characteristic p, the multiplicity has a variant formulation in terms of the so-called lower ramification numbers. These are defined as

in(f) =mult(f^pⁿ(z)) −1.

It is natural to ask what potential sequences of lower ramification numbers are attainable. Sen’s theorem [Sen69] gives insight to this question, and effectively bounds the problem from below, and his theorem says that i_n(f) ≥1+p+ · · · + pⁿ. Polynomials, or in general power series, satisfying the special case of equal- ity are known as minimally ramified, and are investigated further in e.g. [LS97;

LRL16b].

It is known that given a generic polynomial the growth rate of the multiplicity and thus the lower ramification numbers is linear in terms of the iterates, (see e.g.

[NRL20; LRL16b]). Polynomials satisfying this condition are sometimes called pleasantly ramified. In Chapter 2 we provide an example of such a polynomial.

Although, in positive characteristic it is easy to find examples such that the growth rate is exponential and this family of polynomials is the main focus of this thesis. The most straightforward example of such a polynomial is w(z) = z+z^p which has exponential growth rate in terms of the lower ramification numbers (see Theorem 2). In particular, w^pⁿ(z) =z+z^p^pn, i.e. i_n(w) =p^pⁿ−1.

Polynomials such that for no positive integer d,

nlim→_∞

i_n dpⁿ

converges to some finite non-zero limit are sometimes called non-pleasantly ramified polynomials. These are the polynomials of focus for this thesis. Suppose that f is minimally ramified then

nlim→_∞

i_n(f)

dpⁿ = lim

n→_∞ pⁿ⁺¹−1

p−1

dpⁿ = ¹ d.

Hence, minimally ramified polynomials clearly converge to a non-zero limit.

There has been a vast amount of research done concerning lower ramification numbers in general and to list some examples, we have [Sen69; LS97; LS98;

LRL16a; LRL16b; Lin13; Kea92; NRL20; Nor20; KK16].

The results of this thesis concern computing the p-th iterate of a power series in order to find its lower ramification numbers, however, often times the iterates of a power series are far too difficult to compute. As such, many of the aforementioned papers rely on describing the necessary coefficients of the power series as difference equations and thus describing their changes under iteration. By solving these difference equations it is possible to get information about the lower ramification numbers without having to compute the p-th iterate of the power series.

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1.1. Main results 3

Since, we are interested in polynomials, we may do the following relevant ob- servation. Note that

lim sup

n→_∞

mult(_w^pⁿ) deg(w)^pⁿ =1.

However, as mentioned earlier, a typical polynomial exhibits linear growth of lower ramification numbers which thus indicates that this quantity typically is zero. So a natural question to ask is whether there exists polynomials such that this quantity may take on values in the open interval (0, 1). This statement will be further discussed with examples in Chapter 2. In this thesis we give an affirmative answer to this in the case that the field characteristic is 3. In fact, we find an infinite family of polynomials satisfying this property. The proof will be presented in details in the Chapter 4. This limit is of interest due to its relation to the distribution of the periodic points of w.

We conclude this introduction by giving the outline for the thesis and presenting our main results. In Chapter 2 we will present the main motivating examples behind our results. Chapter 3 contains the necessary definitions and theorems used in the proofs. The proofs and main results of the thesis will be contained in Chapter 4. In Chapter 5 we will put the results in a context and discuss them alongside results from [LRL16a; LRL16b; Lin13] from Lindahl and Rivera-Letelier as well as [LS97; LS98] from Laubie and Saïne.

In the next section we present the results of this thesis.

1.1 Main results

The primary result of this thesis are the following theorems and the implications thus provided.

Proposition 1. Let K be a field of characteristic 3. Further let k be a positive integer and P_k(z) ∈K[z] be the polynomial P_k(z) =z(1+z^3k²⁻¹ +z³^k⁻¹). Then for any non-negative integer n we have

P_k³ⁿ(z) =z(1+z^3k

·3n−1

2 +z³^k^·³ⁿ⁻¹). (1.1) Remark. It follows that P_k(z) is an iterate of P`(z) if and only if k = ` ·3^m, for some non-negative integer n. Thus, there are infinitely many integers k such that the polynomial of the form P_k(z) = z(1+z^3k²⁻¹ +z³^k⁻¹) is not an iterate of any of the previous polynomials of such form. Define then, an iterate class as a class of all polynomials [P_n(z)]such that every polynomial in the class is an iterate of P_n(z). Thus, the polynomial family mentioned in Proposition 1 contains an infinite amount of iterate classes and satisfies

nlim→_∞

mult(P_k³ⁿ)

deg(P_k)³ⁿ = lim

n→_∞ 3^k^·³ⁿ−1

2

3^k^·³ⁿ = ¹ 2.

From Proposition 1, it is possible to derive the following consequence, stated as Theorem 1.

Theorem 1. Let K be a field of characteristic 3, and let a∈ {−1, 1}, and b∈K.

Further let Q(z) ∈K[z] be the polynomialQ(z) =az(1+bz+b²z²). Then for any non-negative integer n we have the following

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4 Chapter 1. Introduction

(i) if a= −1, then

Q²^·³ⁿ(z) =z(₁+b³²

·3n−1 2 z³²

·3n−1

2 +b³²^·³ⁿ⁻¹z³²^·³ⁿ⁻¹)_. (1.2) (ii) if a=1, then

Q³ⁿ(z) =z(1+b³³

n−1 2 z³³

n−1

2 +b³³ⁿ⁻¹z³³ⁿ⁻¹). (1.3) The proof relies on Proposition 1, and a properly selected change of coordinates.

A similar proof strategy as used in Theorem 1 is applicable on a different set of polynomials, which are as such included in this thesis in Theorem 2.

Theorem 2. Let K be a field of prime characteristic p. Further let R(z) ∈K[z]be the polynomial R(z) =z(1+z^p^k1⁻¹+z^p^k2⁻¹+...+z^p^km⁻¹), 1<k₁ <k₂<...< k_m. Then for any non-negative integer n we have

R^pⁿ(z) =z(₁+z^p^k1^·^pn⁻¹+z^p^k2^·^pn⁻¹+_...+z^p^km^·^pn⁻¹)_. (1.4) Given the results, it is possible to comment on the lower ramification number of the polynomials considered. In particular,

in(_P_k) = ³^k^·³ⁿ₂⁻¹_, and in(_R) =_p^k¹^·^pⁿ−1.

Proofs of these results can be found in Chapter 4.

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5

2. Motivating examples in

characteristic 3

To give the reader an idea of what happens for different, yet similar, polynomials under iteration we provide in this chapter a few motivating examples with polynomials z+az²+bz³, such that a, b ∈ {0, 1}, where we compute the exact third iterate of three polynomials.

We will show an example of minimally ramified polynomials and two non- pleasantly ramified polynomials which motivates the results of this thesis, specifically Proposition 1 and Theorem 2.

Example (Minimally ramified polynomial z+z²). We begin with computing the third iterate of T(z) =z+z². We have

T²(z) =z+2z²+2z³+z⁴ and

T³(z) =z+3z²+6z³+9z⁴+10z⁵+8z⁶+4z⁷+z⁸. Reducing this modulo 3 yields

T³(z) ≡z+z⁵+2z⁶+z⁷+z⁸ (mod 3).

It follows that the lower ramification number i₁(T)is4, and this is an example of a polynomial which does not have exponential growth in the ramification numbers.

In particular, it is a minimally ramified polynomial.

In order to find all lower ramification numbers for T, i.e. i_n(T), we turn to [LS98, Corollary 1] (see Chapter 5 for a statement of this result). Thus, it is possible to show that

i_n(T) = ³

n+1−1

2 ,

which is further explored in Chapter 5.

Ramification numbers of minimally ramified power series have been extensively researched and as such were not as interesting to our research topic. Hence, we considered polynomials of exponential growth, i.e. non-pleasantly ramified polynomials and their ramification numbers and provide the reader with the following elementary examples along with their iterates in characteristic 3.

Example (Non-pleasantly ramified polynomial z+z³). We begin with computing the third iterate of T(z) =z+z³. We have

T²(z) =z+_2z³+_3z⁵+_3z⁷+z⁹

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6 Chapter 2. Motivating examples in characteristic 3

and

T³(z) =z+3z³+9z⁵+24z⁷+54z⁹+102z¹¹+156z¹³+

+192z¹⁵+189z¹⁷+147z¹⁹+87z²¹+36z²³+9z²⁵+z²⁷. Reducing this modulo 3 yields

T³(z) ≡z+z²⁷ (mod 3).

In order to find lower ramification number for T³ⁿ, i.e. i_n(T), we turn to Corollary 2.1. Letting n=_{1, k}₁=_{1, p} =3 we can show that

i_n(T) =3³ⁿ−1, which makes this polynomial non-pleasantly ramified.

Finally, let us show the motivating example behind Proposition 1, the cubic polynomial T(z) =z+z²+z³.

Example (Non-pleasantly ramified polynomialz+z²+z³). We begin with computing the third iterate of T(z) =z+z²+z³. We have

T²(z) =z+2z²+4z³+6z⁴+8z⁵+8z⁶+6z⁷+3z⁸+z⁹ and

T³(z) =z+3z²+9z³+24z⁴+60z⁵+138z⁶+294z⁷+579z⁸+1053z⁹+ +1767z¹⁰+2739z¹¹+3924z¹²+5196z¹³+6352z¹⁴+7152z¹⁵+ +7389z¹⁶+6969z¹⁷+5961z¹⁸+4587z¹⁹+3144z²⁰+1896z²¹+ +990z²²+438z²³+159z²⁴+45z²⁵+9z²⁶+z²⁷

Reducing this modulo 3 yields

T³(z) ≡z+z¹⁴+z²⁷ (mod 3).

In order to find lower ramification number for T³ⁿ, i.e. i_n(T), we turn to Proposition 1 by which we can show that

i_n(T) = ³

3ⁿ−1 2 , making this polynomial non-pleasantly ramified.

These examples provide a base for our interest in iterates of non-pleasantly ramified polynomials, which will be further studied in Chapter 4.

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7

3. Background

In this chapter we give some important definitions as well as the methods needed for the proofs in Chapter 4.

3.1 Preliminaries

For easier understanding of the research topic we remind ourselves of the most important definitions.

Definition 2 (Ring). A ring is a nonempty set R equipped with two operations ⊕ and ⊗ that satisfy the following conditions.

For all a, b, c∈R:

1. If a∈ R and b∈R, then a⊕b∈ R.

2. a⊕ (b⊕c) = (a⊕b) ⊕c 3. a⊕b=b⊕a

4. There is an element 0_R in R such that a⊕0_R =a,∀a∈ R.

5. For each a ∈R, the equation a⊕x =0_R has a solution in R.

6. If a∈ R, and b∈R, then a⊗b∈ R.

7. a⊗ (b⊗c) = (a⊗b) ⊗c.

8. a⊗ (b⊕c) = (a⊗b) ⊕ (b⊗c)

Definition 3 (Field). A field is a ring (R,⊕,⊗)equipped with additional following axioms

1. a⊗b=b⊗a

2. There is an element 1R, 1R6=0R in R such that a⊗1R= a :∀a ∈R.

3. For each a ∈R^∗, the equation a⊗x=₁_R has a solution in R^∗. 4. For all x, y, z∈ F : x⊗ (y⊕z) = (x⊗y) + (x⊗z)

Since our research concerns polynomials in fields of positive characteristic, it is important to define the following notions.

Definition 4 (Characteristic of a ring). LetK be a field 0_K the zero element of the ring and 1_K the unity of the ring. The characteristic of the ring, denoted char(K) is then defined as the smallest number p such that p⊗1_K =0_K. If char(K)is not finite it is defined as 0.

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8 Chapter 3. Background

The most common fields of characteristic 0 are N, Z, Q, R, C.

Example (Finite field). Fp, i.e. the ring of integers modulo p, is a field of characteristic p.

Example (Infinite field). Fp((T)), i.e. the field of formal Laurent series overFp, is a field of characteristic p.

Definition 5 (Polynomial Ring). Let K be a ring and let x be an indeterminate.

The polynomial ring K[x] is defined as the set of all formal sums anxⁿ+an−1xⁿ⁻¹+...+a₁x+a0=

∑

^aⁱ^xⁱ^,

where each a_i ∈ K.

Given two polynomials, f =_{∑ a}_ixⁱ, g= _{∑ b}_ixⁱ, addition and multiplication are defined as

f+g=

∑

⁽^aⁱ⁺^bⁱ⁾^xⁱ

f g=

∑

i

(

∑

j

a_jb_i−j)xⁱ.

Zero of the polynomial ring is given as the polynomial with zero coefficients.

Finally, we remember the definition of topological conjugacy and dynamically equivalent systems.

Definition 6. Let f : X → X and g : Y → Y be two maps. If there exists a homeomorphism h : X → Y such that h◦ f = g◦h then f and g are said to be topologically conjugate. Two topologically conjugate maps are dynamically equivalent.

In this thesis we will only discuss conjugacy via an affine map ϕ(_z) =αz+β.

3.2 Methods

Large part of the results will be based on a generalized version of the freshman’s dream, i.e. given a field of positive characteristicp, it holds that(a+b)^p= a^p+b^p. In order to generalize that claim we require the following theorems and definitions.

3.2.1 Multinomial Theorem

Multinomial theorem [Hos11] is a generalization of the binomial theorem for three and more variables. It provides a formula for the expansion of a multinomial x₁+x₂+...+x_k raised to a power n.

Theorem 3.1. Let n, k∈_{N and n}_i ∈_{N, 1}≤i≤ k then (x₁+x₂+...+x_k)ⁿ =

∑

n1+n2+...+nk=n n1,n2,...,nk≥0

n

n₁, n2, ..., n_k

x₁ⁿ¹x₂ⁿ²...xⁿ_k^k,

where(_n ⁿ

1,n2,...,nk)is the multinomial coefficient for an ordered k-tuple{n₁, n2, ..., n_k} calculated as

n

n₁, n₂, ..., n_k

= ^n!

n₁!n₂!...n_k!,

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3.2. Methods 9

Example. Multinomial theorem leads us to the following calculation (x+y+z)³=

3 3, 0, 0

x³+

3 2, 1, 0

x²y+

3

2, 0, 1

x²z+

3 1, 2, 0

xy²+ +

3 1, 0, 2

xz²+

3 1, 1, 1

xyz+

3 0, 1, 2

yz²+ +

3 0, 2, 1

y²z+

3 0, 3, 0

y³+

3 0, 0, 3

z³

=x³+y³+z³+3x²y+3x²z+3y²x+3y²z+3z²x+3z²y+6xyz.

3.2.2 Kummer’s Theorem

Kummer’s theorem [Kum52; And73] allows us to find the smallest power of a prime p that divides a binomial coefficient (_mⁿ). However, in order to state said theorem we need the concept of a p-adic valuation which is given below.

Definition 7 (p-adic valuation). Let p be a prime number. Then the p-adic valuation of a positive integer n is the largest exponent ν_p such that p^ν^p divides n. The p-adic valuation can be extended to Q so that the p-adic valuation of a rational number x= ^a_b is defined as ν_p(x) =ν_p(a) −ν_p(b).

Example. By the definition of p-adic valuation, we can calculate that ν₅(50) =2 as50=5²·2. Similarly, ν₇(₂₁⁴) =ν₇(4) −ν₇(21) =0−1= −1.

Theorem 3.1 [Legendre, 1808]. The p-adic valuation of a factorial n! is given by

νp(n!) =

∑

k i=₁

bⁿ pⁱc, where bxc denotes the floor function.

Proof of Legendre’s formula. The amount of times p divides n! is given by the amount of integersk≤ n such that p|k, i.e. bⁿ_pc, plus the amount of integersk≤ n such that p²|k, and so on. This can be written down as

ν_p(n!) =

∑

k i=1

bⁿ pⁱc.

Remark. Legendre’s formula can be written in terms of base-p expansion of n. If we let s_p(n) denote the sum of the digits of n in base-p, then

νp(n!) =

∑

k i=1

bⁿ pⁱc =

∑

k i=1

(n_kp^k⁻¹+ · · · +n_i+1p+n_i) =

∑

k j=1

∑

j i=1

n_jp^j⁻ⁱ

=

∑

k j=1

n_jp^j−1 p−1 =

∑

k j=0

n_jp^j−1 p−1 = ¹

p−1

∑

k j=0

(n_jp^j−n_j)

=ⁿ−s_p(n) p−1 .

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10 Chapter 3. Background

Theorem 3.2 [Kummer, 1852]. Let m, n be two non-negative integers. The p- adic valuation, denoted νp, of the binomial coefficient (_mⁿ) is equal to the number of "carry-overs" when performing the addition of n−m and m, written in base p, i.e.

ν_p(ⁿ m

) = ^S(m) +S(n−m) −S(n)

p−1 ,

where S(m), S(n), S(n−m) denote the sum of digits of m, n and n−m in base p, respectively.

Proof of Kummer’s Theorem. Let n ≥ m be positive integers and consider the p-adic valuation of the binomial coefficient (_mⁿ).

Using the p-adic valuation of a rational number and Legendre’s result for p-adic valuation of factorials, we have

ν_p(ⁿ m

) =ν_p( ^n!

m!(n−m)!) =ν_p(n!) −ν_p(m!) −ν_p((n−m)!)

=ⁿ−s(n) −m+s(m) − (n−m) +s(n−m)

p−1 = ^s(n−m) +s(m) −s(n)

p−1 .

Example. To demonstrate Kummer’s theorem we compute the 5-adic valuation of the binomial coefficientv₅((⁶₂))by showing that(2)₁₀= (2)₅,(6)₁₀ = (11)₅,(4)₁₀= (4)₅. Therefore,S(2) =2, S(6) =2, S(6−2) =4 and

ν5(⁶ 2

) = ^S(2) +S(4) −S(6)

4 = ²+4−2

4 =1.

Kummer’s theorem can be generalized for multinomial coefficients (_m ⁿ

1,m2,...,m_k) as

Theorem 3.2.1 [Generalized Kummer]. Let n, m_i, i∈N be non-negative inte- gers. The p-adic valuation of the multinomial coefficient (_m ⁿ

1,m₂,...,m_k) is equal to ν_p(

n

m₁, m2, ..., m_k

) = ^∑

ki=1S(m_i) −S(n)

p−1 ,

where S(m_i), S(n) denote the sum of digits of m_i and n in base p, respectively.

Example. Applying Kummer’s theorem to the previous example for multinomial theorem leads to the generalized version of freshman’s dream.

Let us consider a field of characteristic p. Then

ν_p(

p

p, 0, ..., 0

) =ν_p(

p

0, ..., p, ..., 0

) =ν_p(

p

0, 0, ..., p

) =⁰+0+1−1 p−1 =0, νp(

p

p₁, p₂, ..., pn

) = ^p¹+_p₂+_...+_p_n−1

p−1 =^p−1

p−1 =1.

Therefore, (x₁+x₂+...+x_n)^p= x₁^p+x₂^p+...+x_n^p in prime characteristic p.

This result naturally extends to any power of the prime p.

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3.2. Methods 11

Remark. Note that for a multinomial coefficient (_p ^p

1,p2,...,pn) = _p ^p!

1!p2!...pn! such that for all integers i, p_i < p we know gcd(_{p!, p}_i_!) =_{1 and}

νp(

p

p₁, p₂, . . . , p_n

) =νp(p!) −νp(p₁!) − · · · −νp(pn) =1−0− · · · −0=1.

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13

4. Results

In this chapter we present the proofs of Proposition 1 and Theorem 1 and 2 along with the corollaries derived from those results.

4.1 Proposition 1

Recall that P_k(z)is defined as z(1+z^3k²⁻¹ +z³^k⁻¹). Throughout the proof, let P(z) =P_k(z)for the sake of simplicity.

In order to prove Proposition 1, it is necessary to prove the following lemmas in the ring K[z] with characteristic 3.

Lemma 1.1.1. For any positive integer m and non-negative integer n we have (z+z^3m²⁺¹ +z³^m)³ⁿ²⁺¹ =z³ⁿ²⁺¹ +2z^3m⁺²³ⁿ +2z^3m

+n+1

2 +z^3m

+n+3m

2 (4.1)

Proof of Lemma 1.1.1. Recall that 3^m+1

2 =1+ (1+3+3²+...+3^m⁻¹). The proof will follow by induction on n. For n=1, we have

(z+z^3m²⁺¹ +z³^m)² =z³⁺²¹ +_2z^3m²⁺³ +_2z^3m

+1+1

2 +z^3m

+1+3m

2 .

Let n=r and

(z+z^3m²⁺¹ +z³^m)^3r²⁺¹ =z^3r²⁺¹ +2z^3m⁺²^3r +2z^3m

+r+1

2 +z^3m

+r+3m

2 .

Then for n=r+1 we have (z+z^3m²⁺¹ +z³^m)^3r

+1+1

2 =(z+z^3m²⁺¹ +z³^m)^3r²⁺¹⁺³^r = (z+z^3m²⁺¹ +z³^m)^3r²⁺¹· (z+z^3m²⁺¹ +z³^m)³^r

=(z^3r²⁺¹ +2z^3m²⁺^3r +2z^3m

+r+1

2 +z^3m

+r+3m

2 )·

(

∑

t1+t2+t3=3^r⁺¹

3^r⁺¹ t₁, t₂, t₃

z^t¹⁺^3m²⁺¹^·^t²⁺³^m^·^t³).

Kummer’s Theorem, mentioned in Chapter 3, allows us to find all of the coefficients in the sum that are divisible by 3. The theorem states that the largest

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14 Chapter 4. Results

power of 3 that divides a trinomial coefficient (_t³^r⁺¹

1,t₂,t₃) is ν3(

3^r⁺¹ t₁, t₂, t₃

) = ^S(t₁) +S(t2) +S(t3) −S(3^r⁺¹)

3−1 = ^S(t₁) +S(t2) +S(t3) −1

2 ,

where S(t_i) denotes the sum of digits of t_i in base 3. We are only interested in coefficients such that ν3((_t³^r⁺¹

1,t₂,t₃)) = 0, i.e. S(t₁) +S(t2) +S(t3) = 1. This equation has solutions only if S(t_i) =1, S(t_j) =S(t_k) =0, where i, j, k= {1, 2, 3}. As S(t_i) = 0 holds if and only if t_i = 0, we can conclude that the summands that will be multiplied with a trinomial coefficient not divisible by 3 are exactly z³^r⁺¹,z^3m⁺^3r²⁺^m⁺¹ and z³^r⁺^m⁺¹.

Now it is possible to reduce the trinomial sum to (z+z^3m²⁺¹ +z³^m)^3r

+1+₁

2 =(z^3r²⁺¹ +2z^3m⁺²^3r +2z^3m

+r+₁

2 +z^3m

+r+_3m

2 )·

(z³^r +z^3r⁺^3r

+m

2 +z³^r⁺^m). Finally, multiplying out the polynomials we get

(z+z^3m²⁺¹ +z³^m)^3r

+1+1 2 =z^3r

+1+1

2 +2z^3m⁺^3r

+1

2 +2z^3m

+r+1+1

2 +z^3m

+r+1+3m

2 +

+_3z²^·^3r⁺^3r

+m+1

2 +_3z^3r⁺²^·^3r

+m+1

2 +_3z²^·^3r⁺^3r

+m+3m

2 +_3z^3r⁺²^·^3r

+m+3m 2

=_z^3r

+1+1

2 +_2z^3m⁺^3r

+1

2 +_2z^3m

+r+1+1

2 +_z^3m

+r+1+3m

2 ,

which proves Lemma 1.1.1.

Lemma 1.1.2. Let P(z) =z+2z³ⁿ²⁺¹ +z³ⁿ+2z³²ⁿ²⁺¹ +2z³²ⁿ²⁺³ⁿ +z³²ⁿ then (P(z))^3m²⁺¹ =z^3m²⁺¹ +z^3m⁺²³ⁿ +z^3m⁺²³²ⁿ +z^3m

+n+1

2 +z^3m

+n+3n

2 +z^3m

+n+32n

2 +z^3m

+2n+1

2 +

+z^3m

+2n+3n

2 +z^3m

+2n+32n

2 ,

and specifically for n=m,

(P(z))³ⁿ²⁺¹ =z³ⁿ²⁺¹ +z³ⁿ+z³²ⁿ²⁺¹ +2z³²ⁿ⁺²³ⁿ +z³²ⁿ+z³³ⁿ²⁺¹ +z³³ⁿ²⁺³ⁿ +z³³ⁿ⁺²³²ⁿ (4.2) Proof of Lemma 1.1.2. It is enough to prove the first part of the lemma asm=n is just a special case of the lemma. We proceed with proof using induction on m.

Form=1,

(P(z))²=(z+2z³ⁿ²⁺¹ +z³ⁿ+2z³²ⁿ²⁺¹ +2z³²ⁿ²⁺³ⁿ +z³²ⁿ)²

=z²+z³⁺²³ⁿ +z³⁺²³²ⁿ +z³¹

+n+1 2 +z³¹

+n+3n

2 +z³¹

+n+32n

2 +z³¹

+2n+1

2 +z³¹

+2n+3n

2 +

+z³¹

+2n+32n 2

Letm=k and

(P(z))^3k²⁺¹ =z^3k²⁺¹ +z^3k⁺²³ⁿ +z^3k⁺²³²ⁿ +z^3k

+n+1 2 +z^3k

+n+3n

2 +z^3k

+n+32n

2 +z^3k

+2n+1

2 +

+z^3k

+2n+3n

2 +z^3k

+2n+32n

2 .

(21)

4.1. Proposition 1 15

Then for m=k+1 and using Kummer’s theorem, we have (P(z))^3k

+1+1

2 = (P(z))^3k²⁺¹⁺³^k = (P(z))^3k²⁺¹(P(z))³^k =

=(z^3k²⁺¹ +z^3k⁺²³ⁿ +z^3k⁺²³²ⁿ +z^3k

+n+1

2 +z^3k

+n+3n

2 +z^3k

+n+32n

2 +z^3k

+2n+1

2 +

+z^3k

+2n+3n

2 + +z^3k

+2n+32n

2 )(z³^k+2z³ⁿ

+k+3k

2 +z³ⁿ⁺^k+2z³²ⁿ

+k+3k

2 +

+_2z³²ⁿ

+k+3n+k

2 +z³²ⁿ⁺^k)

=z^3k

+1+1 2 +z^3k

+1+3n

2 +z^3k

+1+32n

2 +z^3k

+n+1+1

2 +z^3k

+n+1+3n

2 +z^3k

+n+1+32n

2 +

+z^3k

+2n+1+1

2 +z^3k

+2n+1+3n

2 +z^3k

+2n+1+32n

2 ,

which proves the lemma.

Now it is possible to prove Proposition 1 using induction.

Proof of Proposition 1. For k=0, we know that P³⁰(z) =z+z^3m²⁺¹ +z³^m. Assume for n=k

P³^k(z) =z+z^3m^·^3k²⁺¹ +z³^m^·^3k =_{: P}₁(z)_. Let n=k+1, then

P³^k⁺¹(z) =(P³^k)³(z) =P₁³(z) =P₁(P₁(z+z^3m

·3k+1

2 +z³^m^·^3k)) =

=P₁(z+z^3m

·3k+1

2 +z³^m^·^3k+ (z+z^3m

·3k+1

2 +z³^m^·^3k)^3m

·3k+1

2 +

+ (z+z^3m

·3k+1

2 +z³^m^·^3k)³^m^·^3k). By Lemma 1.1.1 this polynomial can be reduced to

P³^k⁺¹(z) =P₁(z+z^3m

·3k+1

2 +z³^m^·^3k +z^3m

·3k+1

2 +2z^3m

·3k+3m·3k

2 +2z^32m

·3k+1

2 +

+z^32m^·^3k²⁺^3m^·^3k + (z+z^3m^·^3k²⁺¹ +z³^m^·^3k)³^m^·^3k)_, which using Kummer’s theorem can be written as

P³^k⁺¹(z) =P₁(z+_2z^3m

·3k+1

2 +z³^m^·^3k +_2z^32m

·3k+1

2 +_2z^32m

·3k+3m·3k

2 +z³^2m^·^3k)_.

Ramification of polynomials

Master’s thesis