On Taylor Series of Zeros with General Base Function

arXiv:2103.07831v1 [math.CV] 14 Mar 2021 nTyo eiso eo ihgnrlbs function base general with zeros of series Taylor On . ee enx xli oeaottestu n eaint prev T to of relation and use set-up the z the including 1.1) about [3], Definition more of explain in those next below We to defined here. techniques (also 2.6 similar [3] use in complex-exponen proofs function” a our “base defined ho term have a We the and and polynomial function). be complex-exponent may base a that (the of function function a sum of a generally, zero a more of or, coefficients series Taylor the for taking oe eetees omlsre xasosmypoieaunifie fi a degree provide of degree. may any polynomials of expansions for polynomials G radicals series of and zeros by formal [9], consider solution Ruffini Nevertheless, [1], general th Abel no of more. objective is work fundamental the there a By is [4], mathematics. polynomial applied a and pure of zeros the Determining Introduction 1 o case special a for rule transformation a series. prove also We 171-181). hs eisuigasse fdffrnileutos trfl [10 series. Taylor Sturmfels of reversion equations. (Gelfand-Kapranov-Zelevin used [7] GKZ-systems differential Herrera using of formulas system [8] same a Mayr the inversion. using Lagrange series using these coefficients series Taylor for las mle n eeaie h nerlt euto trfl “ovn alg (“Solving Sturmfels of coefficie of result these terms integrality about gener in result the more equations integrality is generalizes an series and prove Taylor We gene a implies a Such series. is zero. Puiseux which simple a function a base with a function and morphic polynomial complex-exponent a of surface and 0 6= is ercl ope-xoetplnmas o w ope numb complex two For polynomials. complex-exponent recall we First ayatoshv tde uhfra eis ikln 2 obtain [2] Birkeland series. formal such studied have authors Many epoeafruafrteTyo eiscecet fazr fthe of zero a of coefficients series Taylor the for formula a prove We nti ae,uigTyo eisdrcl,w rv oml (Theo formula a prove we directly, series Taylor using paper, this In z L ob neeeto h imn surface Riemann the of element an be to sprmtie by parametrized is γ edfiecmlxexponentiation complex define we , L = A { hpremti eis.Dsrt ah 1 20)pp. (2000) 210 Math. Discrete series”. -hypergeometric ( ,θ n θ, r, ai DeFranco Mario ac 6 2021 16, March ): r Abstract ∈ R + 1 θ , ∈ ( − ,π π, z γ ] ntecnetoa a by way conventional the in n , L o h oaih.This logarithm. the for ∈ Z } . hs Taylor these f polynomial a polynomial t k)[] [6]. [5], sky) lostudied also obtained ] oswork. ious lomorphic t which nts dformu- ed a to way d roughout a holo- ral e 3.2) rem heorem lthan al ebraic eor ve and , alois sum ers Then for z ∈ L corresponding to (r,θ,n), define zγ by

zγ = eγ ln(r)+iγθ+2πinγ ∈ C.

⇀ ⇀ Deﬁnition 1.1. For an integer d ≥ 1, let a and γ be two d-tuples of complex numbers

⇀ a = (a1,...,ad) ⇀ γ = (γ1,...,γd).

⇀ ⇀ Deﬁne a complex-exponent polynomial p(z; a , γ ) to be a function of the form

p: L → C

d γk z 7→ akz k X=1 which we abbreviate as p(z). Let g(z): C → C be a holomorphic function. We assume that g(z) has a simple zero α 6= 0, and we write its expansion about α as

∞ k g(z)= ck(z − α) (1) k X=1 for some ck ∈ C with c1 6= 0. We call g(z) the “base function”. We ﬁx an element of L that corresponds to α and also denote it by α. Then g(z) also determines a function from a neighborhood V of α in L to C, via the mapping

z 7→ z1 ∈ C for z ∈ V.

⇀ ⇀ Now ﬁx an integer d ≥ 1 and γ ∈ Cd. For an a ∈ Cd, deﬁne the function

⇀ ⇀ f(z; a , γ,g): L → C d γi z 7→ g(z)+ aiz i=1 X ⇀ Assume, for some neighborhood U ⊂ Cd of the origin 0, that there exists a smooth function

⇀ ⇀ φ( a ; γ,g,α): U → L ⇀ ⇀ ⇀ a 7→ φ( a ; γ,g,α)

⇀ that satisﬁes for all a ∈ U

⇀ ⇀ ⇀ ⇀ f((φ( a ; γ,g,α); a , γ,g)) = 0 (2) ⇀ φ(0; γ,g,α)= α. (3)

2 Equations (2) and (3) are suﬃcient to determine the Taylor series coeﬃcients ⇀ ⇀ ⇀ ⇀ of φ( a ; γ,g,α) about 0 in the variables a . ⇀ Let n denote a d-tuple of non-negative integers

⇀ n = (n1,...,nd) and denote d ⇀ Σn = ni. i=1 X ⇀ Let ∂ n denote the partial derivative operator

d ∂ ni ⇀ ∂ n = ( ) , ∂ai i=1 Y and for any function ψ :: U → C (4) denote ⇀ ⇀ ∂(ψ, n)= ∂⇀ψ( a )|⇀. n 0 The Taylor series coeﬃcient formula of Theorem 1.3, [3] (stated as Theorem 2.5 here) is obtained using a base function g(z) of the form

g(z)=1+ bzβ (5) for non-zero b,β ∈ C. As shown in [3], Birkeland [2] and Sturmfels [10] essen- tially find the Taylor series coefficients using the base function (5) with positive integer β and with γi distinct integers not equal to β, though they do not use that terminology or the methods of Taylor series. Their formulas may be ex- pressed in terms of falling factorials. We generalize these formulas in [3] by allowing γi and β to be arbitrary complex numbers (β 6= 0) and show that the factorization is preserved using falling factorials (see Theorem 2.5 here). In this paper, we consider the base function (1) and prove in Theorem 3.2 ⇀ −1 ⇀ that ∂(φ, n) is a polynomial in c1 ,ci, 2 ≤ i ≤ Σn, and that the coefficients of this polynomial factorize as well, using similar falling factorials. Now, the base function (5) is a special case of (1) obtained via

β c = b αβ−k. k k Using this fact combined with Theorem 3.4 gives another proof of Theorem 4.1 in Sturmfels [10]. His proof of Theorem 4.1 uses Hensel’s Lemma, and our proof of Theorem 3.4 counts terms in a derivative. We also note with base function (5), Theorem 3.2 here and Theorem 1.3 ⇀ of [3] give two diﬀerent formulas for the quantity ∂(φ, n). We include in section 5 the objective of ﬁnding a direct proof of this identity.

3 2 Definitions and cited theorems 2.1 Definitions Most of the following definitions occur in [3]. First we present notation for multisets and multiset partitions. For an integer M ≥ 0, we let [1,M] denote

[1,M]= {i ∈ Z: 1 ≤ i ≤ M}.

Deﬁnition 2.1. For a positive integer N, deﬁne an ordered multiset I of [1, d] to be an N-tuple of integers

I = (I(1),...,I(N)) with 1 ≤ I(i) ≤ d. We say that the order |I| is N. We deﬁne the multiplicity #(n, I) of n in I as the number of indices i such that I(i)= n. For a multiset X we also denote the multiplicity of n in X as #(n,X). Let Multiset(d) denote the set of these ordered multisets. For a positive integer k, deﬁne a set partition s of [1,N] with k parts to be a k-tuple s = (s1,...,sk) where si are pairwise disjoint non-empty subsets of [1,N],

k si = [1,N], i=1 [ and min(si) < min(sj ) for i < j.

We also write a set si as an m-tuple

si = (si(1),...,si(m)) where m = |si| and si(j)

J = (J1,...,Jk) where J ∈ Multiset(d) is given by

Ji = (I(si(1)),...,I(si(m))) where m = |si|. Thus the multiset partitions of I with k parts are in bijection with the set partitions in S(|I|, k). Let Parts(I, k) denote the set of such multiset partitions J.

4 ′ Each part Ji of a J ∈ Parts(I, k) is thus a multiset. For J, J ∈ Parts(I, k), we say that J and J ′ are equivalent if there exists a permutation σ of [1, k] such that ′ #(l,Ji) = #(l,Jσ(i)) for each 1 ≤ l ≤ d. We let I(hˆ) denote the ordered multiset obtained from I by removing the element at the h-th index:

I(hˆ) = (I(1),...,I(h − 1), I(h + 1),...,I(N)).

We use the notation N

γm = γI(i) m I i=1 X∈ X where N = |I|. ⇀ For any function ψ( a ) of the form (4) and I ∈ Multiset(d), denote

|I| ∂ ⇀ ∂(ψ, I) = ( )ψ( a )|⇀ ∂a 0 i=1 I(i) Y We also use the falling factorial applied to indeterminates, where “indeterminate” refers to an arbitrary element of some polynomial ring over Z. Definition 2.2. For an integer k ≥ 0 and an indeterminate x, define the falling factorial k (x)k = (x − i + 1). i=1 Y ∞ Let µ denote an infinite sequence (µi)i=1 of non-negative integers µi such that µi = 0 for sufficiently large i. For integer r ≥ 1, let C(r) denote the set of all such µ such that µi = r. i X≥1 Thus C(r) is the set of compositions of r with non-negative parts.

2.2 Cited theorems These results are used here in Theorems 3.2, 3.3, and 4.1. Lemma 2.3. For indeterminates a and b and an integer n ≥ 0,

n n (a + b) = (a) (b) n i i n−i i=0 X and equivalently n a + b a b = . n i n − i i=0 X

5 Proof. This is Lemma 6.1 of [3].

Lemma 2.4. Suppose F (x) is a polynomial of degree m. Then

m+1 k−1 (x − 1) k − 1 F (x)= k−1 (−1)k−1−r F (r + 1). (k − 1)! r k r=0 X=1 X Proof. This is Lemma 2.4 of [3].

⇀ Theorem 2.5. With the above notation and base function (5), for Σn ≥ 1,

⇀ 1+Pd n (γ −1) Σ n −1 d ⇀ α i=1 i i ∂⇀φ( a )|⇀ = − ⇀ (−1+ iβ − niγi) n 0 ′ Σ n g (α) i=1 i=1 Y X where g′(α)= bβαβ−1.

Proof. This is Theorem 1.3 of [3].

Theorem 2.6. Let R be a commutative ring and let δ : R → R be a derivation. For an integer M ≥ 0 and elements fA,fB,fi ∈ R, 1 ≤ i ≤ M,

M |wc|−1 (1) |w| M δ (fA fi)δ (fB fi)= δ (fAfB fi) (6) i∈wc i∈w i=1 w⊂X[1,M] Y Y Y

c (1) where w denotes the complement of w in [1,M]; fA denotes δfA; and in the c −1 (1) case w is empty, δ fA denotes fA. Proof. This is Theorem 4.3 of [3].

Theorem 2.7. For integers 1 ≤ k ≤ N, an indeterminate ν, and N indeterminates xi, 1 ≤ i ≤ N,

k−1 N 1 k − 1 (−1)k−1−r ((r + 1)ν − 1+ x ) (7) (k − 1)! r i N−1 r=0 i=1 X X k k−1 =ν (ν − 1+ xm)|si|−1. (8) i=1 m∈si s∈SX(N,k) Y X Proof. This is Theorem 4.1 of [3].

6 3 Taylor coeﬃcient theorem

Deﬁnition 3.1. Let α be the zero of base function (1). For an x ∈ C and integers r ≥ 0 and a ≥ 1, deﬁne αx x − µ1 F (x, r, a)= (−1) (r + µi)! (a − 1)! r − (i − 1)µi − (a − 1) i≥2 i µ∈XC(r) X≥2 r i µ a µ −( −Pi≥2( −1) i−( −1)) i P α ci × r+1+P µ i≥2 2 µi! c i 1 Y≥2 and also F (x, −1,a)=0. Theorem 3.2. With the above notation and base function (1), and for an I ∈ Multiset(d) with |I|≥ 1,

∂(φ, I)= F ( γm, |I|− 1, 1). (9) m I X∈ Proof. We use induction on |I|. When I = (i), we diﬀerentiate equation (2) ⇀ with respect to ai and evaluate at a = 0 to obtain

γi 0= α + c1∂(φ, I). Solving for ∂(φ, I) proves the base case of |I| = 1. Given an I with |I|≥ 1, suppose equation (9) is true for all I′ with |I′| < |I|. ⇀ Given any function ψ( a ) of the form (4), it follows from the deﬁnitions that

|I| |I|−1 k ⇀ γI(h) −k ∂(f ◦ ψ, I)= (γI(h))kψ( 0 ) ∂(ψ,Ji)) (10) h k i=1 X=1 X=1 J∈Parts(XI(hˆ),k) Y |I| k ⇀ (k) + g (ψ( 0 )) ∂(ψ,Ji). (11) k i=1 X=1 J∈Parts(X I,k) Y Setting ψ to be φ, we obtain

|I| |I|−1 k γI(h) −k 0= (γI(h))kα ∂(φ, Ji) (12) h k i=1 X=1 X=1 J∈Parts(XI(hˆ),k) Y |I| k + k!ck ∂(φ, Ji) (13) k i=1 X=2 J∈Parts(X I,k) Y + c1∂(φ, I). (14) Using the induction hypothesis we may express the sum in the right side of the equation at line 12 as

|I| |I|−1 k γI(h) −k (γI(h))kα F ( xm, |Ji|− 1, 1). h k i=1 m J X=1 X=1 J∈Parts(XI(hˆ),k) Y X∈ i

7 To the sum over J we apply Theorem 3.3 and obtain

|I| |I|−1 γI(h) −k (γI(h))kα F ( xm, |I|− 2, k). h k X=1 X=1 mX∈I(hˆ) Substituting in the deﬁnition of F (x, r, a) and then using

(γI(h))k = (γI(h))(γI(h) − 1)k−1 yields

|I| |I|−1 (γI(h) − 1)k−1 ˆ γi Pi I γi µ1 i∈I(h) −α ∈ γI(h) (−1) (k − 1)! |I|− 2 − (i − 1)µi − (k − 1) h=1 k=1 µ∈C(|I|−2) Pi≥2 X X X −(|I|−1−P (i−1)µ )) µi α i≥2 i c P I µ i . × (| |− 2+ i)! |I|−1+P µ i≥2 2 µi! i c i X≥2 1 Y≥2 Interchanging the sum over k and µ and applying Lemma 2.3 gives

|I| (−1+ i I γi)|I|−2−P (i−1)µi P γi µ1 ∈ i≥2 −α i∈I γI(h) (−1) (|I|− 2 − (i − 1)µi)! h P i≥2 X=1 µ∈CX(|I|−2) and summing over h gives P

∂(φ, I)= F ( xm, |I|− 1, 1). m I X∈ Now combining the result (15) it is suﬃcient to prove

µi (x)|I|−1−P (i−1)µi (|I|− 2+ µi)! c i≥2 µ1 i≥2 i 0= − (−1) |I|−1+P µ (|I|− 2 − (i − 1)µi)! i≥2 i µi! µ C I i≥2 c P i≥2 ∈ X(| |−2) 1 Y P (16)

+ i!ciF ( xm, |I|− 1,i) (17) i m I X≥2 X∈ + c1F ( xm, |I|− 1, 1). (18) m I X∈ νi Take the coeﬃcient of i≥2 ci from each line. Consider the expression

(x) I i Q ν (|I|− 2+ ν )! | |−1−Pi≥2( −1) i |I|−1+P ν i≥2 i (−1) i≥2 i . (19) |I|−1+P νi (|I|− 1 − i≥2(i − 1)νi)! i≥2 c1 P i≥2 νi! P Q 8 The coeﬃcient from line (16) is equal to the expression (19) times

|I|− 1 − (i − 1)νi. (20) i X≥2 The coeﬃcient from the i-th term at line (17) is equal to the expression (19) times i! ν . (21) (i − 1)! i The coeﬃcient from line (18) is equal to the expression (19) times

− (|I|− 1+ νi). (22) i X≥2 Adding the three expressions yields 0:

|I|− 1 − (i − 1)νi + iνi − (|I|− 1+ νi)=0. i i i X≥2 X≥2 X≥2 This completes the proof.

Theorem 3.3. For an integers 1 ≤ a ≤ M and indeterminates xi, 1 ≤ i ≤ M,

a M F ( xm, |si|− 1, 1) = F ( xi,M − 1,a). (23)

i=1 m∈si i=1 s∈SX(M,a) Y X X Proof. We use induction on M. Equation (23) is true when M = 1. Assume it is true for all values less than or equal to some M ≥ 1. We note the equation is true when a = 1 for any M. For 1 ≤ a ≤ M, we have by the induction hypothesis

a+1 F ( xm, |si|− 1, 1)

i=1 m∈si s∈S(MX+1,a+1) Y X c = F ( xm, |w |− 1,a)F ( xm, |w|− 1, 1) (24) m∈wc m∈w w⊂[1,M+1]X,M+1∈w X X where w is the set of the set partition s that contains M + 1. For elements ν ∈ C(M) and µ ∈ C(j) for some j, we say µ ≤1 ν if

µi ≤ νi for i ≥ 2

µ1 ≤ ν1 − 1.

′ If µ ≤1 ν, denote µ ∈ C(M − 1 − j) by

′ µi = νi − µi for i ≥ 2 ′ µ1 = ν1 − 1 − µ1.

9 Fix an element ν ∈ C(M). Then the coeﬃcient of

νi ci (25) i Y≥2 in the right side of equation (24) is

PM+1 x −α i=1 i (26) (a − 1)! w⊂[1,M+1]X,M+1∈w µ≤1ν,µX∈C(|w|−1) c ′ −(|w |−1−Pi≥2(i−1)µi−(a−1)) µ′ m wc xm c ′ α 1 ∈ w µ (−1) c ′ (| |− 1+ i)! |wc|+P µ′  |w |− 1 − (i − 1)µ − (a − 1) i≥2 i ′  i≥2 i i c µ ! P X≥2 1 i≥2 i  (27)  P Q

−(|w|−1−P (i−1)µi) xm α i≥2 µ1 m∈w w µ . × (−1) (| |− 1+ i)! |w|+P µ  |w|− 1 − (i − 1)µi i≥2 i  i≥2 i c µ ! P X≥2 1 i≥2 i  (28) P Q

For an integer b ≥ 0, the coeﬃcient of (xM+1)b at line (28) is

PM+1 x −(M−P (i−1)ν −a) (−1)ν1+1α i=1 i i≥2 i (29) M+1+Pi≥2 νi c1 c ′ 1 c x (|u |− 1+ µi)! × m∈u m i≥2 a b uc a i µ′ µ′ ( − 1)! ! | |− 1 − ( − 1) − i≥2( − 1) i i≥2Pi! u⊂X[1,M] µ≤1ν,µX∈C(|u|) P P (30)Q x (|u| + µi)! × m∈u m i≥2 (31) |u|− b − (i − 1)µ µ ! P i≥2 i iP≥2 i P Q where we have set u = w \{M +1}. The coefficient of the product of (xM+1)b M+1 and the monomial (25) in F ( i=1 ,M,a + 1) is equal to the product of the expression at line (29) and P M (M + νi)! x i≥2 i=1 i . (32) a!b! ν ! M − a − b − (i − 1)ν Pi≥2 i P i≥2 i Therefore it is sufficientQ to prove that sum atP lines (30) and (31) is equal to expression (32). We prove this next by comparing the coefficients of the variables xi. Fix integers l and ni ≥ 0 such that l ≤ M. Consider the monomial

l ni xi . (33) i=1 Y

10 The coeﬃcient of monomial (33) in the sum at lines (30) and (31) is

M 1 (34) (a − 1)!b! j=0 v⊂X[1,l] X µ≤1ν,µX∈C(j) ′ M−j−1−(a−1)−Pi (i−1)µi ≥2 (M − j − 1+ µ′ )! Pi∈vc ni ( i∈vc ni)! i≥2 i ′ ′ (M − j − 1 − (a − 1) − (i − 1)µ )! c ni! µ ! i≥2 i Pi∈v i≥2 Pi (35) P Q Q j−b−Pi≥2(i−1)µi P ni ( ni)! (j + µi)! × i∈v i∈v i≥2 (36) (j − b − (i − 1)µi)! ni! µi! i≥2 Pi∈v Pi≥2 M − l × P Q Q (37) j − |v| M − l where we have set j = |u| and used the fact that there are ways to j − |v| choose a set u ∈ [1,M] such that v ⊂ u and vc ⊂ uc. ′ Now each term in the above sum is independent of ν1,µ1, and µ1. We thus re-arrange the summation at line (34) to be

M (38) µ ν j=0 v⊂X[1,l] X˜≤˜ X whereν ˜ is the ﬁxed sequence (νi)i≥2;µ ˜ is any sequence of non-negative integers (˜µi)i≥2 with µ˜i ≤ νi for i ≥ 2 and ′ µ˜i = νi − µ˜i for i ≥ 2. The coeﬃcient of term (33) in expression (32) is

M−a−b−P (i−1)νi i≥2 l (M + νi)! l i≥2 Pi=1 ni ( i=1 ni)! l . (39) a!b! νi! (M − a − b − (i − 1) νi)! Pi≥2 i≥2 Pi=1 ni! ! We equate expressionQ (39) with the sumP beginning at lineQ (34) (using the or-

11 dering (38)) and simplify to obtain

M (40) µ ν j=0 v⊂X[1,l] X˜≤˜ X M−j−1−(a−1)−P (i−1)˜µ′ ′ i≥2 i (M − j − 1+ µ˜i)! P c ni a i≥2 i∈v ( n )! (M − j − 1 − (|vc|− 1))! (M − j − 1 − (a − 1) − (i − 1)˜µ′ )! i P i≥2 i i∈vc X P (41) j−b−P (i−1)µi j µ i≥2 ( + i≥2 ˜i)! Pi∈v ni × ( ni)! (42) (j − |v|)! (j − b − (i − 1)˜µi)! P i≥2 i∈v ′ X (˜µi +˜µi)! P × ′ (43) (˜µi)!(˜µ )! i i Y≥2 M−a−b−P (i−1)ν i≥2 i l (M + νi)! l i≥2 Pi=1 ni = ( ni)!. (44) (M − l)! (M − a − b − (i − 1)νi)! P i≥2 i=1 X We claim the above equation is trueP for any integer M ≥ l. Thus multiply both sides above the above equation by tM−l and sum over M ≥ l. The claim is thus equivalent to the equality of power series

(45) µ ν v⊂X[1,l] X˜≤˜ ′ c ′ d P µ˜ +|v |−1 a−1 P c n i µ˜ ( ) i≥2 i (at (− ln(1 − t)) i∈v i (t ) i ) (46) dt i Y≥2 d P µ˜i+|v| b P n i µ˜ × ( ) i≥2 (t (− ln(1 − t)) i∈v i (t ) i ) (47) dt i Y≥2 ′ (˜µi +˜µi)! × ′ (48) (˜µi)!(˜µ )! i i Y≥2 l d P νi+l a b P n i ν = ( ) i≥2 (t t (− ln(1 − t)) i=1 i (t ) i ). (49) dt i Y≥2 This equality follows from Theorem 2.6 using the ring R of power series in t; δ diﬀerentiation with respect to t; and

a fA = t b fB = t ni fi = (− ln(1 − t)) for 1 ≤ i ≤ l, such that for each integer h ≥ 2 there are exactly νh indices i with l +1 ≤ i ≤ l + ≥2 νh and f = th. P i Here we have used the fact that the number (48) counts the number of subsets of c [1,l+ ≥2 νh] that contain v, do not contain v , and contain exactlyµ ˜h numbers P 12 h i such that fi = t , for each h ≥ 2. This proves the claim and completes the proof.

Theorem 3.4. Suppose γi ∈ Z for 1 ≤ i ≤ d. Then ⇀ ⇀ ⇀ ∂(φ, n) n · γ −1 −1 α Z α ,c ,c ,c ,...,c ⇀ . d ∈ [ 1 2 3 Σ n ] (50) i=1 ni! ⇀ Proof. We use inductionQ on Σn. From Theorem 3.2, statement (50) is true ⇀ ⇀ ⇀ when Σn = 1. Assume it is true for all n with Σn ≤ m for some m ≥ 1. Given ⇀ ⇀ a n with Σn = m +1, ﬁx an r such that nr ≥ 1. Consider

γr ⇀ ∂ n (arφ ). (51) ⇀ ⇀ After setting a to 0, expression (51) is equal to

⇀ Σ n −1 k γr −k nr (γr)kα ∂(φ, Ji) (52) k i=1 X=1 J∈Parts(X I,k) Y where I is the ordered multiset satisfying I(h) ≤ I(h + 1), and with r occurring with multiplicity nr − 1 and i occurring with multiplicity ni for all other i. Now ﬁx a k and J. Suppose that there are v distinct sets out of the k sets of J. Call these sets J˜1,..., J˜v with J˜i appearing bi times in J. The number of J ′ ∈ Parts(I, k) equivalent to J is then d 1 #(i,J)! v k . bi! i=1 i=1 l=1 #(i,Jl)! Y ′Q d Collect these terms for J equivalent toQ J and divide by i=1 ni!. The total coeﬃcient is then Q γr −k d k nr(γr)kα 1 #(i,J)! d v k ∂(φ, Ji) bi! nr i=1 #(i,J)! i=1 i=1 l=1 #(i,Jl)! i=1 Y Y k γQ Q Q ∂(φ, J ) = r multinomial((b )v ) l k i i=1 d l i=1 #(i,Jl)! Y=1 where have used v Q bi = k. i=1 X By the induction hypothesis

∂(φ, Jl) P γm −1 −1 m∈Jl Z d ∈ α [α ,c1 ,c2,...,c|Jl|], i=1 #(i,Jl)! γ and r is anQ integer by the assumption that γ is an integer. k r Next, consider a ﬁxed k-th term at line (13) where I is the ordered multiset satisfying I(h) ≤ I(h + 1), and with ni = #(i, I). Collecting terms for all equivalent J and applying similar reasoning above yields an element in the set at line (50), with k! taking the role of (γr)k. This completes the proof.

13 4 A transformation rule

Now we present deﬁnitions for Theorem 4.1. Using base function (5), we rename its zero α by α1, and we denote the ⇀ ⇀ corresponding zero φ( a ; γ,g,α1) of f(z) by

⇀ ⇀ φ( a ; γ,b,β1).

For a non-zero β2 ∈ C, we denote α2 by

1 β2 α2 = α1 . (53)

⇀ Let M(d) denote the set of d-tuples n

⇀ d n = (ni)i=1

⇀ of non-negative integers ni such that Σn ≥ 1. ⇀ ⇀ ⇀ For n ∈ M(d), let V (k, n) denote the set of k × d arrays v

⇀ v = (vi,j ) for 1 ≤ i ≤ d and 1 ≤ j ≤ k where k vi,j = ni j=1 X and vj ∈ M(d) where d vj = (vi,j )i=1. ⇀ k Let perm( v ) denote the number of permutations of the multiset (vj )j=1. Let ⇀ ⇀ w(n, γ,β) denote

d ⇀ ⇀ −1 −1 w n, γ,β β β n γ ⇀ . ( ) = ( 1 − 1+ 1 i i)Σ n −1 i=1 X ⇀ ⇀ Theorem 4.1. With the above notation, as Taylor series about a = 0

⇀ ⇀ 1 ⇀ ⇀ β φ( a ; γ,b,β1) 2 = φ( a ; β2 γ,b,β2β1).

Proof. We have by Theorem 2.5

d P ni(γi−1) d d ⇀ i=1 ⇀ ni ⇀ ⇀ Σ n α1 Σ n −1 −1 −1 ai φ a γ,b,β α β β β n γ ⇀ ( ; 1)= 1 1+ (−1) ⇀ 1 ( 1 − 1+ 1 i i)Σ n −1  ′ Σ n ni!  ⇀ g (α) i=1 i=1 n ∈XM(d) X Y   d ni d ⇀ ⇀ −1 Pi=1 niγi ai = α1 1+ β1 α1 w(n, γ,β) .  ni!  ⇀ i=1 n ∈XM(d) Y  

14 Therefore

1 β2 1 d ni ⇀ ⇀ 1 d n γ ⇀ ⇀ a β β2 −1 Pi=1 i i i φ( a ; γ,b,β1) 2 = α1 1+ β1 α1 w(n, γ,β)  ni!  ⇀ i=1 n ∈XM(d) Y   ∞ −1 d ni d ⇀ ⇀ −k β2 Pi=1 niγi ai k = α2 1+ β1 ( α1 w(n, γ,β) )  k ni!  k ⇀ i=1 X=1 n ∈XM(d) Y   ⇀ ⇀ by the binomial theorem, using the fact that a is suﬃciently close to 0 . d mi ⇀ a For an m ∈ M(d), the coeﬃcient of i in the above sum is mi! i=1 Y

d ⇀ Pi miγi Σm −1 d k α =1 β ⇀ ⇀ α 1 β−k 2 (perm(v) mult((v )k )) w(v , γ,β) 2 d 1 k i,j j=1 j i=1 mi! k ⇀ i=1 j=1 X=1 v∈VX(k,m) Y Y Q (54) Let I ∈ Multiset(d) be the ordered multiset in which #(i, I)= mi and ordered ⇀ so that I(j) ≤ I(j +1). For J ∈ Parts(I, k), let w(J, γ,β) denote

k d ⇀ −1 −1 w(J, γ,β)= (β − 1+ β γi)|Jh|−1. h i J Y=1 X∈ h Then we may write expression (54) as

d ⇀ Pi=1 miγi Σm α1 −1 −1 k−1 ⇀ α2 d (β2 − 1)k−1(β ) w(J, γ,β). (55) β1β2 i=1 mi! k X=1 J∈Parts(X I,k) By TheoremQ 2.7, we have

k−1 ⇀ 1 k − 1 (β−1)k−1 w(J, γ,β)= (−1)k−1−r ((r+1)β−1−1+ γ ) (k − 1)! r 1 i |I|−1 r=0 i I J∈Parts(X I,k) X X∈

Let ck denote the right side of the above equation. By Lemma 2.4, we have expression (55) is equal to

d Pi=1 miγi d α1 −1 −1 α β β β m γ ⇀ . 2 d (( 1 2) − 1+ 1 i i)Σm−1 β1β2 i=1 mi! i=1 X Q d ami Using equation (53), we see that this is the coeﬃcient of i in the series mi! i=1 ⇀ ⇀ Y for φ( a ; β2 γ,b,β2β1). This completes the proof.

15 5 Further work

• Find a combinatorial meaning of the integer coefficients of the ci terms using trees. • Find a combinatorial meaning of the factored coefficients in [3] when a g(z)=1+ i+d zd. ai • Generalize NRS using these combinatorial meanings. • Directly prove the factorization of coefficients in [3] from the formulas here. • Generalize Theorem 4.1 for complex-exponent polynomials g(z) with mul- tiple terms. • Use the method of Lagrange or a modification to construct iterated radical formulas for polynomial zeros and relate them to Taylor series.

References

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