Conservative Polynomials

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Conservative Polynomials Casey Briggs Summer Research Project November 2009 - January 2010 Supervisor: Associate Professor Finnur L´arusson School of Mathematical Sciences University of Adelaide

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Contents 1 Introduction 1.1 Examples of Conservative Polynomials . . . . . . . . . . . . . . . .

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2 Literature Review

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3 General form of a normalised polynomial

7

4 Finiteness of the set Cn

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5 Equivalent conservative polynomials 6 Conservative polynomials of 6.1 Degree 1 . . . . . . . . . . 6.2 Degree 2 . . . . . . . . . . 6.3 Degree 3 . . . . . . . . . . 6.4 Degree 4 . . . . . . . . . . 6.5 Degree 5 . . . . . . . . . . 6.6 Higher degrees . . . . . . .

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7 Conclusion

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A Computer Code

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2

1

Introduction

We investigate a concept that was first introduced by Smale [1981], that of the conservative polynomial. Definition Let f (z) be a polynomial with complex coefficients. We say that f is normalised if it satisfies f (0) = 0 and is monic. Definition Let f (z) be a polynomial with complex coefficients. If f 0 (θ) = 0 implies f (θ) = θ, then we say that f is conservative. Put simply, f is conservative if all of its critical points are fixed points. We concentrate on the set Cn of those conservative polynomials of degree n that are normalised. There is an important result, that this set is finite for each n ∈ N, detailed in Section 4. We also consider a special class of conservative polynomial, the real conservative polynomial. Definition A polynomial f with real coefficients is real conservative if it is conservative and all of its critical points are real (have no imaginary part). Note that if f is a normalised conservative polynomial with real critical points, then it must be real conservative. This paper serves as a brief investigation into the topic. In Section 2 we survey the literature and outline key known results on conservative polynomials. We then find the general form of a normalised polynomial given its critical points, and in Section 4 use this to prove a result from this survey, namely, that there are only finitely many normalised conservative polynomials at a given degree. Beginnings are made to a catalogue of normalised conservative polynomials in Section 6. Methods of doing this are developed, and with more time and more computing power, this catalogue could be continued. We investigate various means of classifying normalised conservative polynomials, and show two equivalence relations on the set Cn in Section 5.

1.1

Examples of Conservative Polynomials

The monomial g(z) = z n is conservative for all n ∈ N. It has only one critical point, namely, 0.

3

n z is conservative for all n ≥ 2. Let us The normalised polynomial f (z) = z n + n−1 verify this. Firstly, we have d n n 1 n z + z = nz n−1 + = 0 ⇐⇒ z n−1 = − . dz n−1 n−1 n−1 1 Let α be a critical point of f . It follows then that αn−1 = − n−1 . We want to verify that f (α) = α.

f (α) = αn +

nα n−1

α nα + n−1 n−1 = α. =−

Thus f is conservative. Note that the critical points are solutions α of αn−1 = −

1 6= 0, n−1

1 and the non-zero number − n−1 has n − 1 distinct (n − 1)th roots. So a conservative polynomial of this form has n − 1 distinct critical points.

4

2

Literature Review

There is a relatively small literature on the topic. Much ground has been covered by Smale [1981], Yagzhev [1987], Tischler [1989], Kostrikin [1984] and Pakovich [2008]. An overview of some of the known results follows. Smale [1981] was the first to introduce the concept of the conservative polynomial, in a discussion of complexity theory. He did not explore the topic in any great depth, but hypothesised that there are only finitely many normalised conservative polynomials at each degree. Kostrikin [1984] outlines a method for generating new conservative polynomials from existing ones, in the form of the following proposition. Proposition 2.1 If f is a normalised conservative polynomial, c is a fixed point of f , and n−1 = 1, then g(z) = −1 [f ( z + c) − c] is normalised and conservative. Proof First observe that g(0) = −1 [f (c) − c] = 0 since c is a fixed point of f . Also, the coefficient of z n in g(z) is −1 n = n−1 = 1, so g is normalised. Let Θf be the set of all critical points of f . Since g 0 (z) = f 0 ( z + c), Θg = { −1 (θ − c)|θ ∈ Θf }. So g( −1 (θ − c)) = −1 [f ( ( −1 (θ − c) + c)) − c] = −1 [f (θ) − c] = −1 (θ − c), i.e. g is conservative. 2 When = 1 we then just have “translationsâ€? of the conservative polynomial by a fixed point, and when c = 0 (0 is always a fixed point of a normalised polynomial), we have “rotationsâ€?. This concept will be explored in greater detail in Sections 5 and 6. Kostrikin also considers a number of special cases, in the hope of finding a complete description of the conservative polynomials, and states several hypotheses and results, including the following proposition. different normalised conservative polyProposition 2.2 There are precisely n n−1 2 nomials with a derivative of the form n(z − θ0 )n−1−r (z − θ1 )r ,

1≤r≤

n−1 . 2

That is, we know exactly how many normalised conservative polynomials there are at each degree with 2 distinct critical points. 5

Kostrikin stresses that his results are only a starting point in the study of conservative polynomials. Indeed, many of the subsequent papers reference, and make use of his results. It is mentioned that a widening of the empirical data on conservative polynomials using computers would be useful; an attempt at this is made by the beginnings of a catalogue of normalised conservative polynomials in Section 6. He also hypothesises that the set of normalised conservative polynomials is finite for each n ∈ N. This was proven by Yagzhev [1987], and a detailed exposition of this proof is given in Section 4. Tischler [1989] includes a discussion of normalised conservative polynomials in his paper about the critical points of complex polynomials. His approach involves showing a bijective correspondence between conservative polynomials and certain types of alternating trees. This creates an analogue of conservative polynomials in the discrete, combinatorial object of an alternating tree. Two of these alternating trees are then defined to be equivalent if there is an orientation preserving homeomorphism of C taking one to the other which preserves the orientation of the edges. Two polynomials are defined to be equivalent if they differ by conjugation by a degree one polynomial. That is, two polynomials f and g are related if there is a degree one polynomial A such that g = A−1 ◦ f ◦ A. This is an equivalence relation. Tischler [1989] uses this and subsequent results to show the following result, improving on the result shown by Yagzhev [1987]. Theorem 2.3 The cardinality of Cn is 2n−2 for all n ∈ N. n−1 This is clearly a useful result. We now know how many normalised conservative polynomials there are at each degree. Pakovich [2008] takes the results of Tischler [1989] as a launchpad for his own investigation in the topic. He investigates in greater detail the correspondence between conservative polynomials and bicoloured trees (trees with vertices coloured in two colours in such a way that any edge connects vertices of different colours).

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3

General form of a normalised polynomial

Any normalised polynomial can be specified uniquely by its critical points, counted with multiplicities. If P (z) is a normalised polynomial of degree n with critical points α1 , . . . , αn−1 , then P 0 (z) = n(z − α1 ) · · · (z − αn−1 ). Then n−1

n−1 i−1

n n−2 X X n n−1 X αi + αi αj − · · · + nα1 . . . αn−1 (−1)n−1 z z z P (z) = z − n−1 n−2 i=1 i=1 j=1   n

=

n X

n n−p zp   p (−1) p=1

X i1 ,...,in−1 ∈{0,1} i1 +···+in−1 +1=p

 1−i α11−i1 · · · αn−1n−1  

Pulling the embedded sum to the front gives the expression P (z) = n

X i1 ,...,in−1 ∈{0,1}

z

i1 +···+in−1 +1 (−1)

n−i1 −···−in−1 −1

i1 + · · · + in−1 + 1

1−i

α11−i1 · · · αn−1n−1

(1)

We can use this form of a normalised polynomial to find examples of conservative polynomials.

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4

Finiteness of the set Cn

Yagzhev [1987] concentrates on one particular result in his paper, of which a detailed proof will be provided in this section. Theorem 4.1 The set Cn of normalised conservative polynomials of degree n is finite for each natural number n. The proof of Theorem 4.1 appeared originally in Yagzhev [1987], and was corrected in Yagzhev [1988], what follows is a more detailed exposition of the proof. Proof Let g(z) be a normalised polynomial of degree n + 1 with critical points α1 , . . . , αn . Note that the case where n = 0 is trivial, since there is only one normalised polynomial (let alone normalised conservative polynomials) of degree 1. So assume n ≥ 1. Then g is conservative if and only if g(αs ) = αs for all s ∈ {1, . . . , n}. That is (from Equation 1 with n replaced by n + 1), (n + 1)

X i1 ,...,in ∈{0,1}

αsi1 +···+in +1

(−1)n−i1 −···−in 1−i1 α . . . αn1−in = αs i1 + · · · + in + 1 1

(2)

Order the critical points α1 , . . . , αn of g such that αi 6= 0 for i ≤ m, and αj = 0 for j > m, for some m ∈ {1, . . . , n}. That is, order the critical points such that all critical points at zero appear at the end of the list. The summand in (2) is zero for all instances where ik = 0, k > m. So we can assume ik = 1 if k > m, and simplify the sum. Then 1−im i1 +···+im +n−m+1 α11−i1 . . . αn1−in αsi1 +···+in +1 = α11−i1 . . . αm αs m 1 1 i1 +···+im +n−m+1 Y αi = i1 · · · im α s αm α1 i=1 i1 im m Y αs αs n−m+1 = ··· αs αi α1 αm i=1

for all s ∈ {1, . . . , m}. Also, i1 + · · · + in + 1 = i1 + · · · + im + n − m + 1, and

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(−1)n−i1 −···−in = (−1)i1 +···+im −m . So from equation (2) we obtain X

(n+1)

i1 ,...,im ∈{0,1}

(−1)i1 +···+im i1 + · · · + im + n − m + 1

αs α1

i1

...

αs αm

im

= (−1)m αsm−n

m Y

!−1 αi

i=1

(3) for all s ∈ {1, . . . , m} as the condition on g being conservative. Now, let |αt | = min |αi |. 1≤i≤m

(4)

Then from relation 3, in the case where s = t,

−1

−1 m m

Y Y

n−m

αi = (−1)m αtn−m αi

αt

i=1 i=1

i1 im

i +···+i m X 1

αt αt (−1)

··· = (n + 1)

i + · · · + i + n − m + 1 α α m 1 m

i1 ,...,im ∈{0,1} 1 # "

i1

im X

αt

αt

1

≤ (n + 1) · · ·

|i1 + · · · + im + n − m + 1| α1 αm i1 ,...,im ∈{0,1} X ≤ (n + 1) 1 i1 ,...,im ∈{0,1}

= (n + 1)|{0, 1}m | = (n + 1)2m ≤ (n + 1)2n since m ≤ n ≤ 2n 2n (proof by induction) = 4n . Thus

m

n−m Y

αi ≥ 4−n .

αt

(5)

i=1

Then from (4), |αsn−m

Qm

i=1

αi | ≥ 4−n for each s ∈ {1, . . . , m}.

Definition A closed algebraic subset X of Cn is the set of zeroes of a finite set of polynomials f1 , . . . , fm , that is, the set of all x = (x1 , . . . , xn ) ∈ Cn such that fi (x) = 0, i = 1, . . . , m.

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So the possible critical points (α1 , . . . , αn ) form a closed algebraic set S in Cn , by (2). We need the following lemma (stated here without proof), which follows from Proposition 2.31 of Mumford [1976]. Lemma 4.2 If the range of a rational function, defined on a closed algebraic set in Cm , is contained in the complement of an infinite subset of C, then it is finite. Q From (5), we have that for each s = 1, . . . , m, the range of αsn−m m i=1 αi is contained in C\{x ∈ C : |x| ≤ 4−n }. QThat is, the image of the polynomial map S → C, where (α1 , . . . , αn ) 7→ αsn−m m i=1 αi , omits a disc about 0. So by Lemma Q m n−m 4.2, αs i=1 αi can take only a finite number of values for each s ∈ {1, . . . , m}. This means that m Y s=1

αsn−m

m Y

! αi

=

i=1

m Y

!n−m+1 αi

i=1

can take only finitely many values, since it is the product of functions that can each only take finitely many values. Q th roots Then m i=1 αi can take only finitely many values, namely the (n − m + 1) of a function which takes finitely many values. Suppose m < n. That is, suppose that 0 is a critical point of the normalised conservative polynomial g. Then αsn−m can be expressed as a quotient of non-zero functions Q α αsn−m m n−m Qm i=1 i . αs = i=1 αi That is, αsn−m is the quotient of functions that can take only finitely many values. So αsn−m can only take finitely many values, implying that each αs can take only finitely many values. This shows that there are finitely many normalised conservative polynomials of a given degree with 0 as a critical point. Now let g be a normalised conservative polynomial of degree n + 1 with m = n. That is, g has no critical point at 0. Let α1 , . . . , αn be the critical points of g. Define a new polynomial h(z) = g(z + α1 ) − α1 . Then h is conservative, and has critical points at 0, α2 − α1 , . . . , αn − α1 (see Proposition 2.1). Now h has a critical point at 0, so α2 − α1 , . . . , αn − α1 can only take a finite number of values. Let B be the finite set of these possible values.

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Write αj = α1 + (αj − α1 ) = α1 + bj , for some bj ∈ B, 2 ≤ j ≤ n. Plug αj into relation (2) for s = 1 to get the polynomial equation α1 = (n+1)

X

α1i1 +···+in +1

i1 ,...,in ∈{0,1}

(−1)n−i1 −···−in 1−i1 (α1 +b2 )1−i2 · · · (α1 +bn )1−in . α i1 + · · · + in + 1 1

The leading term of the right hand side is X

(n + 1)α1n+1

i1 ,...,in ∈{0,1}

(−1)n−i1 −···−in . i1 + · · · + in + 1

Now n ≥ 1, so to show that the polynomial equation has only finitely many solutions for α1 , we need to show that the polynomial is not the zero polynomial. To do this it suffices to show that the coefficient of the leading term is nonzero. That is, we need to show that (n + 1)

X i1 ,...,in ∈{0,1}

(−1)n−i1 −···−in 6= 0. i1 + · · · + in + 1

By (1), this term is P (1), where P is the normalised polynomial of degree n + 1 with critical points all equal to zero, that is, P 0 (z) = (n + 1)(z − 1)n . So P (z) = (z − 1)n+1 − (−1)n+1 , and the sum is P (1) = (−1)n 6= 0. So there are finitely many values that α1 can take. This shows that there are finitely many normalised conservative polynomials of a given degree. 2

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5

Equivalent conservative polynomials

We already have a method of generating new conservative polynomials from old ones, described in Proposition 2.1. First, let us consider the special case where = 1. Definition Let f ∈ Cn . Define f ≥ g if g(z) = f (z + c) − c, for some fixed point c of f . Clearly g ∈ Cn if f ≥ g (see the proof of Proposition 2.1) Proposition 5.1 The binary relation ≥ is an equivalence relation. Proof Reflexivity: Clearly f ≥ f , since 0 is a fixed point of all normalised polynomials. Symmetry: Suppose f ≥ g. Then g(z) = f (z + c) − c, where c is a fixed point of f. Now, g 0 (z) = f 0 (z + c) = 0 if and only if z + c = θi for θi ∈ Θf . So the critical points of g are Θg = {θi − c : θi ∈ Θf } Observe that g(−c) = f (−c+c)−c = −c, so −c is a fixed point of g. Then consider the polynomial g(z − c) + c, which has critical points only where z − c = θi − c, that is, only where z = θi . Given that g(z −c)+c has the same critical points as f (z), so they must be equal (a normalised conservative polynomial is uniquely determined by its critical points, counted with multiplicities). This shows that g ≥ f . Transitivity: Let f, g, h ∈ Cn , f ≥ g and g ≥ h. Then g(z) = f (z + c) − c h(z) = g(z + d) − d

for some fixed point c of f and for some fixed point d of f.

So h(z) = f (z + d + c) − c − d. Note that Θg = {θi − c : θi ∈ Θf } and Θh = {θi − d : θi ∈ Θg } = {θi − (c + d) : θi ∈ Θf } Observe that h(−(c + d)) = g(−c − d + d) − d = f (−c + c) − d − c = −(c + d), so −(c + d) is a fixed point of h. Consider the polynomial h(z − (c + d)) + (c + d), which has critical points only where z − (c + d) = θi − (c + d), that is, only where z = θi . 12

Given that h(z − (c + d)) + (c + d) has the same critical points as f (z), they must be equal. This shows that f ≥ h. 2 We can now look at equivalence classes of conservative polynomials, in which each class contains polynomials of the same “shapeâ€?. Now let us consider the generalisation of this equivalence relation. Definition Let f, g ∈ Cn . Define f âˆź g if g(z) = −1 [f ( z + c) − c], for c some fixed point of f and some ∈ C with n−1 = 1 Then by Proposition 2.1, g ∈ Cn . Proposition 5.2 The binary relation âˆź is an equivalence relation. Proof Reflexivity: Let = 1. f is a normalised polynomial, so 0 is a fixed point of it. So let c = 0. This shows that f âˆź f . Symmetry: Suppose f âˆź g. That is, g(z) = −1 [f ( z + c) − c] ∴ f ( z + c) = g(z) + c ∴ f (w) = [g( −1 w − −1 c) − (− −1 c)] where w = z + c. Note that ( −1 )n−1 = 1−1 = 1, so we just need − −1 c to be a fixed point of g. g(− −1 c) = −1 [f (0) − c] = − −1 c, since 0 is a fixed point of the normalised polynomial f . This shows that g âˆź f . Transitivity: Suppose that f âˆź g and g âˆź h. That is, g(z) = −1 1 [f ( 1 z + c) − c] h(z) = −1 2 [g( 2 z + d) − d] where c is a fixed point of f , d is a fixed point of g, and 1n−1 = 2n−1 = 1 Then we can express h in terms of f : −1 h(z) = −1 2 [ 1 [f ( 1 2 z + 1 d + c) − c] − d] −1 −1 −1 −1 = −1 2 1 f ( 1 2 z + c + 1 d) − 2 1 c − 2 d = ( 2 1 )−1 [f ( 1 2 z + (c + 1 d)) − (c + 1 d)].

Note that ( 2 1 )n−1 = n−1 n−1 = 1. 1 2 13

−1 −1 By symmetry of â€˜âˆźâ€™, we already have that f (z) = 1 [g( −1 1 z − 1 c) − (− 1 c)], so −1 −1 f (c + 1 d) = 1 [g( −1 1 (c + 1 d) − 1 c) + 1 c] = 1 [g(d) + −1 1 c] = 1 d + c, since d is a fixed point of g.

So c + 1 d is a fixed point of f , and thus we have shown that f âˆź h. 2

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6

Conservative polynomials of small degree

There is an obvious method for finding normalised conservative polynomials. We already have the general form of a normalised conservative polynomial P of degree n (see Section 1). So to find all conservative polynomials, we simply need to solve simultaneously the equations P (αi ) = αi , for all i = 1, . . . , n − 1. At small degrees, this is a trivial task. By approximately degree 6 however, the complexity of the system of equations we generate is too great for a desktop computer to solve. At this point, we investigate other methods of generating conservative polynomials, including making use of the equivalence classes in Section 5. The catalogue of normalised conservative polynomials was calculated by hand up to degree 3. Catalogues at higher degrees were found using computer assistance. The Matlab functions written can be found in Appendix A. The catalogue is sorted in two ways, by the equivalence relations ‘≡’ and ‘∼’ as defined in Section 5. Since ≡ is a special case of ∼, each equivalence class of ≡ is contained entirely within an equivalence class of ∼. In each table, separate equivalence classes under ≡ (that is, sets of polynomials of the same shape) are separated by a single horizontal line, and separate equivalence classes under ∼ are separated by triple horizontal lines. We list the polynomials by their critical points, in polar form. The critical point αj is given by the two parameters rj = |αj | and θj = arg αj in the interval (−π, π]. The real conservative polynomials are shown in italics.

6.1

Degree 1

The only normalised conservative polynomial of degree 1 is P (z) = z. This is trivially conservative, since it has no critical points at all. P (z) is a real conservative polynomial.

6.2

Degree 2

There are 2 normalised conservative polynomials of degree 2. They are z 2 and z 2 + 2z, with a critical point at 0 and −1 respectively. They are both real, and in the same equivalence class under both ≡ and ∼.

15

Table 1: Conservative Polynomials of Degree 3 r1 0 1.000000000000000 1.000000000000000 0 0 0.707106781186548

6.3

θ1 0 0 3.141592653589793 0 0 1.570796326794897

r2 0 1.000000000000000 1.000000000000000 1.414213562373095 1.414213562373095 0.707106781186548

θ2 0 0 3.141592653589793 1.570796326794897 -1.570796326794897 -1.570796326794897

Degree 3

There are 6 normalised conservative polynomials of degree 3, of which 3 are real. They are shown in Table 1.

6.4

Degree 4

There are 20 normalised conservative polynomials of degree 4, of which 6 are real. They are shown in Table 2.

6.5

Degree 5

There are 70 normalised conservative polynomials of degree 5, of which 8 are real. They are shown in Table 3.

16

17

r1 0 1.000000000000000 1.000000000000000 1.000000000000000 0 2.068558771549385 0.335190512296296 0 0 0.335190512296296 0 2.068558771549385 0 0 2.068558771549385 0.335190512296296 0 0 0 0.693361274350635

θ1 0 3.141592653589793 1.047197551196598 -1.047197551196598 0 3.141592653589793 3.141592653589793 0 0 1.047197551196598 0 1.047197551196598 0 0 -1.047197551196598 -1.047197551196598 0 0 0 -1.047197551196598

r2 0 1.000000000000000 1.000000000000000 1.000000000000000 0 0.626309201241976 1.107059058011113 1.442249570307408 0 1.107059058011113 1.442249570307408 0.626309201241976 1.442249570307408 0 0.626309201241976 1.107059058011113 1.200936955176003 1.200936955176003 1.200936955176003 0.693361274350635

θ2 0 3.141592653589793 1.047197551196598 -1.047197551196598 0 3.141592653589793 0 0 0 -2.094395102393196 -2.094395102393196 1.047197551196598 2.094395102393196 0 -1.047197551196598 2.094395102393196 -0.523598775598299 2.617993877991494 -2.617993877991494 1.047197551196598

r3 0 1.000000000000000 1.000000000000000 1.000000000000000 1.442249570307408 0.626309201241976 1.107059058011113 1.442249570307408 1.442249570307408 1.107059058011113 1.442249570307408 0.626309201241976 1.442249570307408 1.442249570307408 0.626309201241976 1.107059058011113 1.200936955176003 1.200936955176003 1.200936955176003 0.693361274350635

Table 2: All Conservative Polynomials of Degree 4 θ3 0 3.141592653589793 1.047197551196598 -1.047197551196598 3.141592653589793 3.141592653589793 0 0 1.047197551196598 -2.094395102393196 -2.094395102393196 1.047197551196598 2.094395102393196 -1.047197551196598 -1.047197551196598 2.094395102393196 0.523598775598299 1.570796326794897 -1.570796326794897 3.141592653589793

18

θ1 0 0 3.141592653589793 -1.570796326794897 1.570796326794897

0 -2.971094892721540 -0.785398163397448 1.400298565926643 -0.785398163397448 0 2.971094892721540 -1.400298565926643 0.785398163397448 0.785398163397448 0 -2.356194490192345 1.741294087663150 -0.170497760868253 -2.356194490192345 0 2.356194490192345 0.170497760868253 -1.741294087663150 2.356194490192345

0 3.141592653589793 3.141592653589793 3.141592653589793 0 0 -1.570796326794897 1.570796326794896 -1.570796326794898 -1.570796326794895

0 -2.356194490192347 0.785398163397448 -2.356194490192344 0.785398163397449 0 2.356194490192347 -0.785398163397448 2.356194490192344 -0.785398163397449

r1 0 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000

0 0.758587860546141 1.228777428265950 0.758587860546141 1.414213562373095 0 0.758587860546141 0.758587860546141 1.228777428265950 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095

0 1.977895382902313 1.565084580073286 0.782542290036644 0.412810802829025 0 1.977895382902313 0.412810802829025 0.782542290036644 1.565084580073286

0 1.106681919700323 0.903602003609844 0.903602003609844 1.106681919700323 0 1.106681919700323 0.903602003609844 0.903602003609844 1.106681919700323

0 1.106681919700323 0.903602003609844 0.903602003609844 1.106681919700323 0 1.106681919700323 0.903602003609844 0.903602003609844 1.106681919700323

0 1.977895382902313 1.565084580073286 0.782542290036644 0.412810802829025 0 1.977895382902313 0.412810802829025 0.782542290036644 1.565084580073286

0 0.758587860546141 1.228777428265950 0.758587860546141 1.414213562373095 0 0.758587860546141 0.758587860546141 1.228777428265950 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095

0 -2.356194490192347 0.785398163397448 -2.356194490192344 0.785398163397449 0 2.356194490192347 -0.785398163397448 2.356194490192344 -0.785398163397449

0 3.141592653589793 3.141592653589793 3.141592653589793 0 0 -1.570796326794897 1.570796326794896 -1.570796326794898 -1.570796326794895

0 -2.971094892721540 -0.785398163397448 1.400298565926643 -0.785398163397448 0 2.971094892721540 -1.400298565926643 0.785398163397448 0.785398163397448 0 -2.356194490192345 1.741294087663150 -0.170497760868253 -2.356194490192345 0 2.356194490192345 0.170497760868253 -1.741294087663150 2.356194490192345

θ2 0 0 3.141592653589793 -1.570796326794897 1.570796326794897

1.106681919700322 2.213363839400645 0.203079916090478 2.010283923310165 0 1.106681919700322 2.213363839400645 0.203079916090478 2.010283923310165 0

1.565084580073287 0.412810802829026 0 0.782542290036643 1.977895382902312 1.565084580073287 0.412810802829026 1.977895382902312 0.782542290036643 0

0 0.758587860546141 1.228777428265950 0.758587860546141 1.414213562373095 0 0.758587860546141 0.758587860546141 1.228777428265950 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095 0 1.228777428265950 0.758587860546141 0.758587860546141 1.414213562373095

r3 0 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000

-2.356194490192345 -2.356194490192346 -2.356194490192344 -2.356194490192344 0 2.356194490192345 2.356194490192346 2.356194490192344 2.356194490192344 0

0 3.141592653589793 0 0 0 1.570796326794897 -1.570796326794899 1.570796326794897 1.570796326794899 0.344481318536683

0 -2.971094892721540 -0.785398163397448 1.400298565926643 -0.785398163397448 0 2.971094892721540 -1.400298565926643 0.785398163397448 0.785398163397448 0 -2.356194490192345 1.741294087663150 -0.170497760868253 -2.356194490192345 0 2.356194490192345 0.170497760868253 -1.741294087663150 2.356194490192345

θ3 0 0 3.141592653589793 -1.570796326794897 1.570796326794897

Table 3: All Conservative Polynomials of Degree 5 r2 0 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000

1.106681919700322 0 2.010283923310165 0.203079916090478 2.213363839400645 1.106681919700322 0 2.010283923310165 0.203079916090478 2.213363839400645

1.565084580073287 0.412810802829026 0 0.782542290036643 1.977895382902312 1.565084580073287 0.412810802829026 1.977895382902312 0.782542290036643 0

1.414213562373095 1.952744029912679 0.185436134107145 1.952744029912679 0 1.414213562373095 1.952744029912679 1.952744029912679 0.185436134107145 0 1.414213562373095 0.185436134107145 1.952744029912679 1.952744029912679 0 1.414213562373095 0.185436134107145 1.952744029912679 1.952744029912679 0

r4 0 1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000

0.785398163397448 3.063866843609940 0.785398163397448 0.785398163397439 0.785398163397448 -0.785398163397448 0 -0.785398163397448 -0.785398163397439 -0.785398163397448

0 3.141592653589793 0 0 0 1.570796326794897 -1.570796326794899 1.570796326794897 1.570796326794899 0.344481318536683

2.356194490192345 2.679092949565149 2.356194490192345 2.033296030819542 0 -2.356194490192345 -2.679092949565149 -2.033296030819542 -2.356194490192345 0 0.785398163397448 0.785398163397448 1.108296622770252 0.462499704024645 0 -0.785398163397448 -0.785398163397448 -0.462499704024645 -1.108296622770252 0

θ4 0 0 3.141592653589793 -1.570796326794897 1.570796326794897

19

0 -2.848749881861219 2.848749881861219 3.141592653589793 0 0 -0.292842771728575 0.292842771728575 3.141592653589793 0 0 -1.570796326794897 1.570796326794897 -1.863639098523472 -1.277953555066321 0 -1.570796326794897 1.570796326794897 1.863639098523472 1.277953555066321

-2.356194490192345 -1.570796326794897 0 -2.356194490192345 3.141592653589793

0 1.414213562373093 1.414213562373093 1.122455255285951 0.445452054899617 0 1.414213562373095 1.414213562373095 0.445452054899617 1.122455255285947 0 1.122455255285947 0.445452054899617 1.414213562373095 1.414213562373095 0 0.445452054899617 1.122455255285947 1.414213562373095 1.414213562373095

0.707106781186548 1.000000000000000 0 1.414213562373095 1.000000000000000

0.707106781186548 0 1.000000000000000 1.000000000000000 1.414213562373095

0 1.414213562373093 1.414213562373093 1.122455255285951 0.445452054899617 0 1.414213562373095 1.414213562373095 0.445452054899617 1.122455255285947 0 1.122455255285947 0.445452054899617 1.414213562373095 1.414213562373095 0 0.445452054899617 1.122455255285947 1.414213562373095 1.414213562373095 2.356194490192345 0 1.570796326794897 3.141592653589793 2.356194490192345

0 -2.848749881861219 2.848749881861219 3.141592653589793 0 0 -0.292842771728575 0.292842771728575 3.141592653589793 0 0 -1.570796326794897 1.570796326794897 -1.863639098523472 -1.277953555066321 0 -1.570796326794897 1.570796326794897 1.863639098523472 1.277953555066321 0.707106781186548 1.414213562373095 1.000000000000000 1.000000000000000 0

1.414213562373095 0.816496580927723 0 0.469342731533017 1.845187632832261 1.414213562373095 0.816496580927726 0 1.845187632832261 0.469342731533019 1.414213562373095 0.469342731533019 1.845187632832261 0.816496580927726 0 1.414213562373095 1.845187632832261 0.469342731533019 0.816496580927726 0 -0.785398163397448 -0.785398163397448 0 -1.570796326794897 0

-0.292842771728575 -1.570796326794895 0 -1.054857211987804 -0.223096343078522 -2.848749881861218 -1.570796326794897 0 -2.918496310511271 -2.086735441601994 1.863639098523472 2.625653538782696 1.793892669873419 3.141592653589793 0 -1.863639098523472 -1.793892669873419 -2.625653538782696 3.141592653589793 0

0.707106781186548 1.000000000000000 1.414213562373095 0 1.000000000000000

1.414213562373095 0 0.816496580927723 0.469342731533017 1.845187632832261 1.414213562373095 0 0.816496580927726 1.845187632832261 0.469342731533019 1.414213562373095 0.469342731533019 1.845187632832261 0 0.816496580927726 1.414213562373095 1.845187632832261 0.469342731533019 0 0.816496580927726

0.785398163397448 0 0.785398163397448 0 1.570796326794897

0.292842771728575 0 1.570796326794895 1.054857211987804 0.223096343078522 2.848749881861218 0 1.570796326794897 2.918496310511271 2.086735441601994 1.277953555066321 0.515939114807098 1.347699983716374 0 0 -1.277953555066321 -1.347699983716374 -0.515939114807098 0 0

6.6

Higher degrees

Beyond degree 5, the computations involved in solving the necessary set of simultaneous equations are too intensive for a desktop computer to achieve in a realistic time frame. There are 252 normalised conservative polynomials of degree 6, and this number increases rapidly from this point forth. Other methods can be employed to generate some or all of the normalised conservative polynomials at each degree. For example, we can simplify the equations by looking for conservative polynomials with a critical point of multiplicity at least 2. This reduces the number of variables, and the number of equations by 1. Further simplifications can be made to find some of the conservative polynomials at each degree. We can also make use of the equivalence relations defined in Section 5. The transformation from one normalised conservative polynomial to another one is a process with relatively low complexity. So we could generate many more normalised conservative polynomials from one that is known to be normalised and conservative.

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7

Conclusion

This report presents a far from complete investigation of the normalised conservative polynomial. Further avenues for exploration include, but are not necessarily limited to: • The special case of the real normalised conservative polynomial. How many of these are there at each degree? It appears from the empirical data in this report that there are relatively few of these, and the proportion of them to all normalised conservative polynomials drops as the degree is raised. A real conservative polynomial has the property that all of its critical points must lie on the diagonal x = y on the Cartesian plane. How does this restrict the general form of a conservative polynomial? • The bijective correspondence between classes of conservative polynomials and classes of bicoloured plane trees. What results can be derived from this relationship? • Conservative polynomials that are not necessarily normalised. Is there a general form or easy method of finding these? Can any general results or properties be shown? • The motivation behind the initial definition of conservative polynomials. How are they useful?

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A

Computer Code

polynomialGenerator.m function f = p o l y n o m i a l G e n e r a t o r ( n , c r i t i c a l P o i n t s ) % Gen erates and o u t p u t s t h e n o r m a l i s e d p o l y n o m i a l o f d e g r e e n with % c r i t i c a l p o i n t s c o n t a i n e d i n t h e 1∗( n−1) v e c t o r criticalPoints ( Critical % P o i n t s g i v e n i n c a r t e s i a n form ) . % % Written by Casey B r i g g s 12/01/2010 syms f g z ; M = combn ( [ 1 0 ] , n−1) ; dimensionsOfM = s i z e (M) ; f = 0; for i = 1 : dimensionsOfM ( 1 ) power = sum(M( i , : ) ) + 1 ; g = z ˆ ( power ) ∗(( −1) ˆ ( n−power ) ) / ( power ) ; for j = 1 : n−1 g = g ∗ ( ( c r i t i c a l P o i n t s ( j ) ) ) ˆ(1−M( i , j ) ) ; end f = f + g; end f = n∗ f ; end

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polynomialEvaluator.m function f = p o l y n o m i a l E v a l u a t o r ( n , c r i t i c a l P o i n t s , X) % Evaluates the normalised polynomial of degree n with % c r i t i c a l p o i n t s c o n t a i n e d i n t h e 1∗( n−1) v e c t o r criticalPoints ( Critical % P o i n t s g i v e n i n c a r t e s i a n form ) a t t h e p o i n t X. % % Written by Casey B r i g g s 12/01/2010

syms f g z ; M = combn ( [ 1 0 ] , n−1) ; dimensionsOfM = s i z e (M) ; f = 0; for i = 1 : dimensionsOfM ( 1 ) power = sum(M( i , : ) ) + 1 ; g = Xˆ ( power ) ∗(( −1) ˆ ( n−power ) ) / ( power ) ; for j = 1 : n−1 g = g ∗ ( ( c r i t i c a l P o i n t s ( j ) ) ) ˆ(1−M( i , j ) ) ; end f = f + g; end

f = n∗ f ; end

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fixedEqnGenerator.m function f = f i x e d E q n G e n e r a t o r ( n , c r i t i c a l P o i n t s ) % Gen erates and o u t p u t s t h e e q u a t i o n f ( z ) − z t h a t must be solved for 0 , to % find the f i x e d points of a normalised Conservative Polynomial f , o f d e g r e e n w i t h % c r i t i c a l p o i n t s c o n t a i n e d i n t h e 1∗( n−1) v e c t o r criticalPoints ( Critical % P o i n t s g i v e n i n c a r t e s i a n form ) . % % Written by Casey B r i g g s 12/01/2010 syms f g z ; M = combn ( [ 1 0 ] , n−1) ; dimensionsOfM = s i z e (M) ; f = 0; for i = 1 : dimensionsOfM ( 1 ) power = sum(M( i , : ) ) + 1 ; g = z ˆ ( power ) ∗(( −1) ˆ ( n−power ) ) / ( power ) ; for j = 1 : n−1 g = g ∗ ( ( c r i t i c a l P o i n t s ( j ) ) ) ˆ(1−M( i , j ) ) ; end f = f + g; end f = n∗ f − z ; end

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translation.m function t r a n s l a t i o n s = t r a n s l a t i o n ( n , c r i t ) % Takes a p o l y n o m i a l f ( z ) ( s p e c i f i e d by i t s c r i t i c a l p o i n t s ) of degree n % and t r a n s f o r m s i t i n t o a l l p o l y n o m i a l s o f t h e form % g(z) = f (z + c) − c % where c i s a f i x e d p o i n t o f f % g ( z ) t h e n has c r i t i c a l p o i n t s a i − c , where a i are t h e c r i t i c a l points % of f % Written By Casey B r i g g s 12/01/2010. translations = [ ] ; f = fixedEqnGenerator (n , c r i t ) ; fixedPoints = solve ( f ) ; fixed = [ ] ; critMatrix = c r i t ; for j = 1 : n−1 fixed = [ fixed , fixedPoints ] ; critMatrix = [ critMatrix ; c r i t ] ; end translations = critMatrix − fixed ; end

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equivalent.m function l i s t = e q u i v a l e n t ( n , c r i t ) % Takes a p o l y n o m i a l f ( z ) ( s p e c i f i e d by i t s c r i t i c a l p o i n t s ) of degree n % and g e n e r a t e s a l l members o f t h e e q u i v a l e n c e c l a s s g e n e r a t e d by i t , under % the relation % g ( z ) = e p s ˆ( −1) [ f ( e p s ∗ z + c ) − c ] % where c i s a f i x e d p o i n t o f f and e p s ˆ( n−1) = 1 % g ( z ) t h e n has c r i t i c a l p o i n t s e p s ˆ( −1) [ a i − c ] , where a i are t h e c r i t i c a l p o i n t s % of f % Written by Casey B r i g g s 12/01/2010. syms m q = mˆ ( n−1) ; q = q − 1; allowableEps = solve (q) ; allowableEps = allowableEps ’ ; list = []; translations = [ ] ; f = fixedEqnGenerator (n , c r i t ) ; fixedPoints = solve ( f ) ; fixed = [ ] ; critMatrix = c r i t ; for j = 1 : n−1 fixed = [ fixed , fixedPoints ] ; critMatrix = [ critMatrix ; c r i t ] ; end translations = critMatrix − fixed ; for k = 1 : n for t = 1 : length ( a l l o w a b l e E p s ) l i s t = [ l i s t ; a l l o w a b l e E p s ( t ) ˆ( −1) ∗ t r a n s l a t i o n s ( k ,:) ]; end end end 26

cartesian2polar.m function polar = c a r t e s i a n 2 p o l a r ( s o l n ) % C o n v e r t s complex numbers i n C a r t e s i a n form ( x+i y ) , i n t o P o l a r form % ( norm∗ exp ( i ∗ argument ) ) . % I n p u t : n∗m m a t r i x o f complex numbers i n c a r t e s i a n form . % Output : n∗2m m a t r i x o f complex numbers i n p o l a r form . Each p a i r o f % columns c o r r e s p o n d s t o one number , g i v i n g t h e two p a r a m e t e r s o f t h e number . % eg . The f i r s t two columns o f t h e o u t p u t c o r r e s p o n d t o t h e norm and % argument ( r e s p e c t i v e l y ) o f t h e numbers i n t h e f i r s t column o f t h e i n p u t . % % Written By Casey B r i g g s 12/01/2010.

dim = s i z e ( s o l n ) ; polar = [ ] ; for i = 1 : dim ( 1 ) for j = 1 : dim ( 2 ) x = double ( s o l n ( i , j ) ) ; polar ( i , 2∗ j −1) = norm( x ) ; polar ( i , 2∗ j ) = angle ( x ) ; end end

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polar2cartesian.m function c a r t e s i a n = p o l a r 2 c a r t e s i a n ( matrix ) % C o n v e r t s complex numbers i n C a r t e s i a n form ( x+i y ) , i n t o P o l a r form % ( norm∗ exp ( i ∗ argument ) ) . % I n p u t : n∗2m m a t r i x o f complex numbers i n p o l a r form . Each p a i r o f % columns c o r r e s p o n d s t o one number , g i v i n g t h e two p a r a m e t e r s o f t h e number . % eg . The f i r s t two columns o f t h e o u t p u t c o r r e s p o n d t o t h e norm and % argument ( r e s p e c t i v e l y ) o f t h e numbers i n t h e f i r s t column o f t h e o u t p u t . % Output : n∗m m a t r i x o f complex numbers i n c a r t e s i a n form . % % Written By Casey B r i g g s 12/01/2010. dim = s i z e ( matrix ) ; cartesian = [ ] ; for k = 1 : dim ( 1 ) temp = [ ] ; for j = 1 : dim ( 2 ) /2 temp = [ temp , matrix ( k , 2 ∗ j −1) ∗( cos ( matrix ( k , 2 ∗ j ) ) + sqrt ( −1)∗ sin ( matrix ( k , 2 ∗ j ) ) ) ] ; end c a r t e s i a n = [ c a r t e s i a n ; temp ] ; end

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testConservativity.m function t e s t C o n s e r v a t i v i t y ( n , c r i t ) % Tests whether normalised polynomials of degree n with c r i t i c a l points ( in % C a r t e s i a n form ) i n t h e m a t r i x % crit is conservative . % Each row o f c r i t r e p r e s e n t s t h e c r i t i c a l p o i n t s o f a new conservative % polynomial % % Written By Casey B r i g g s 12/01/2010 eps = 0 . 0 0 0 0 1 ; sum = 0 ; a = size ( c r i t ) ; n u m b er o f p o l y n om i a l s = a ( 1 ) ; for k = 1 : nu m b e r o f p ol y n o m i a l s eval = [ ] ; for m = 1 : n−1 eval = [ eval , p o l y n o m i a l E v a l u a t o r ( n , c r i t ( k , : ) , c r i t ( k ,m) ) ] ; end error = eval − c r i t ( k , : ) ; sumError = 0 ; for p = 1 : length ( error ) sumError = sumError + abs ( error ( p ) ) ; end sumError i f ( sumError < eps ) disp ( ’ c o n s e r v a t i v e ’ ) else disp ( ’ not c o n s e r v a t i v e ’ ) end end 29

combn.m function [M, IND ] = combn (V,N) % COMBN − a l l c o m b i n a t i o n s o f e l e m e n t s % M = COMBN(V,N) r e t u r n s a l l c o m b i n a t i o n s o f N e l e m e n t s of the elements in % v e c t o r V. M has t h e s i z e ( l e n g t h (V) . ˆN)−by−N. % % [M, I ] = COMBN(V,N) a l s o r e t u r n s t h e i n d e x m a t r i x I so t h a t M = V( I ) . % % V can be an a r r a y o f numbers , c e l l s or s t r i n g s . % % Example : % M = COMBN( [ 0 1 ] , 3 ) r e t u r n s t h e 8−by−3 m a t r i x : % 0 0 0 % 0 0 1 % 0 1 0 % 0 1 1 % ... % 1 1 1 % % A l l e l e m e n t s i n V are r e g a r d e d as unique , so M = COMBN ([2 2] ,3) returns % a 8−by−3 m a t r i x w i t h a l l e l e m e n t s e q u a l t o 2 . % % NB Matrix s i z e s i n c r e a s e s e x p o n e n t i a l l y a t r a t e ( nˆN) ∗N . % % See a l s o PERMS, NCHOOSEK % and ALLCOMB and PERMPOS on t h e F i l e Exchange % % % %

f o r Matlab R13 , R14 v e r s i o n 4 . 0 (may 2008) ( c ) Jos van der Geest email : jos@jasen . nl

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References A. I. Kostrikin. Conservative polynomials. Studies in Algebra, pages 115–129, 1984. David Mumford. Algebraic Geometry I. Springer-Verlag Berlin Heidelberg New York, 1976. ¯ Fedor Pakovich. Conservative polynomials and yet another action of Gal(Q/Q) on plane trees. J. Thor. Nombres Bordeaux, 20(1):205–218, 2008. Steve Smale. The fundamental theorem of algebra and complexity theory. Bulletin of the American Mathematical Society, 4(1), 1981. David Tischler. Critical points and values of complex polynomials. Journal of Complexity, 5:438 – 456, 1989. A. V. Yagzhev. Finiteness of the set of conservative polynomials of given degree. Matematicheskie Zametki, 41(2):148–151, 1987. A. V. Yagzhev. Letter to the editor. Matematicheskie Zametki, 48(2):285, 1988.

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