Uniqueness of entire and meromorphic functions

sharing finite sets

by Bernard Shiffman^{1}

1. Introduction.

In 1921, G. Pólya [10] showed that an entire function is uniquely determined by the inverse images, counting multiplicities, of 3 distinct non-omitted values. A few years later, R. Nevanlinna [7] showed that a meromorphic function on the complex line C is uniquely determined by the inverse images, ignoring multiplicities, of 5 distinct values. Specifically, if two nonconstant meromorphic functions f, g : C → C ∪ {∞} “share” 5 distinct values a_{1}, …, a_{5}, that is, if f^{−1}(a_{j}) = g^{−1}(a_{j}) for 1 ≤ j ≤ 5 (ignoring multiplicities), then f = g. When multiplicities are considered, then 4 points suffice (with one exceptional situation) by the argument of Pólya [10]; see [8, p. 122]. A related problem, asked by F. Gross [3], is to find a finite set S so that an entire function is determined by the single pre-image, counting multiplicities of S. We call such a set a uniqueness set for entire functions counting multiplicities. Answers to this question and the analogous question for meromorphic functions on C were given by H.-X. Yi [12] and P. Li and C.-C. Yang [4], [5], who investigated the zero sets of polynomials P of the form

P (t) = t^{n} + at^{n−m} + 1 ,

where n > m ≥ 1 and a is chosen so that P has n distinct roots. Yi [12] showed that the zero set of P is a uniqueness set for entire functions counting multiplicities if n ≥ 2m + 5, m ≥ 5 and (m, n) = 1; this result was extended to m ≥ 1 by Li-Yang [4], yielding a 7 point uniqueness set. Furthermore, Li and Yang [5] showed that the zero set of P is a uniqueness set for meromorphic functions counting multiplicities if m ≥ 2, n ≥ 2m+11, and (m, n) = 1. For related results on uniqueness sets for polynomials, see [9], [13].

In this paper we show that the zero set of P is a uniqueness set ignoring multiplicities for suitable values of n, m. The precise results are given as parts (a) and (c) of Theorem 1 below. We also give a slight improvement (part (b) of Theorem 1) of the uniqueness result in [5] stated above.

We use the usual notation f^{−1}(S) for the inverse image (without multiplicities) of a finite set S by the map f, and we let f^{∗}(S) be the pull-back by f of the divisor S, or the “inverse image with multiplicities.” Specifically, for a ∈ C, we write f^{∗}{a} = ∑ m_{j}{x_{j}} (finite or countably infinite sum), where the x_{j} are the zeros of f − a and m_{j} ∈ ZZ^{+} is the multiplicity of the zero at {x_{j}}; for a finite set S = {a_{1}, …, a_{n}}, we let f^{∗}(S) = f^{∗}{a_{1}} + ⋯ + f^{∗}{a_{n}}.

1Research partially supported by National Science Foundation Grant No. DMS-9500491.

The following theorem combines our results on uniqueness sets for entire and meromorphic functions ignoring multiplicities—parts (a) and (c)—with the results on uniqueness sets counting multiplicities—parts (b) and (d)—mentioned above. Part (b) of Theorem 1 was also obtained independently by Mues [6] and by Shirosaki [11] (and was given by Li and Yang [5] for n ≥ 2m + 11). Part (d) is due to Li and Yang [4]. Our proof gives a unified treatment of all four cases.

Theorem 1. Suppose that n, m are positive integers with n > m and let

S = {t ∈ C : t^{n} + at^{n−m} + 1 = 0} ,

where a ∈ C ^{\} {0} is chosen so that S contains n distinct points. Suppose that f, g are nonconstant meromorphic functions on C. Then f = cg for a constant c with c^{n} = c^{m} = 1, and in particular f = g when (m, n) = 1, if one of the following holds:

a) n ≥ 2m + 17, m ≥ 2 and f^{−1}(S) = g^{−1}(S);

b) n ≥ 2m + 9, m ≥ 2 and f^{∗}S = g^{∗}S;

c) n ≥ 2m + 9, m ≥ 1, f, g are entire and f^{−1}(S) = g^{−1}(S);

d) n ≥ 2m + 5, m ≥ 1, f, g are entire and f^{∗}S = g^{∗}S.

In particular, if we take the smallest m, n in Theorem 1, we obtain:

a) a 21 point uniqueness set for meromorphic functions without multiplicities,

b) a 13 point uniqueness set for meromorphic functions with multiplicities,

c) an 11 point uniqueness set for entire functions without multiplicities,

d) a 7 point uniqueness set for entire functions with multiplicities.

There still remains much to be answered concerning this problem. For instance, are there sets with fewer elements than specified above? More generally, we conjecture that for a generic finite set S ⊂ C containing sufficiently many points,

f^{∗}(S) = g^{∗}(S) ⇒ f = g .

Let P ∈ C[t] be the polynomial whose roots are the points of S. In fact, we show in Section 4 that P (f) = P (g) ⇒ f = g, for a generic polynomial P ∈ C[t] of degree at least 5. Thus, this conjecture reduces to showing that f^{∗}(S) = g^{∗}(S) ⇒ P (f) = P (g) for generic P of sufficiently high degree.

2. Review of Nevanlinna theory.

We let w = [w_{1}, …, w_{k+1}] denote a point in complex projective k-space, CIP^{k}. Suppose that h : C → CIP^{k} is a holomorphic curve, and let h̃ = (h̃_{1}, …, h̃_{k+1}) : C → C^{k+1} ^{\} {0} be a (regular) representation of h, where the h̃_{j} are entire functions. The growth function T (h) is given by

T (h)(r) = T (h, r) = \frac{1}{2π} ∫_{0}^{2π} log ∥h̃(re^{iθ})∥dθ .  (1)

We let Div f = f^{∗}(0) denote the divisor of a holomorphic function f on a domain in C, as described in Section 1. Suppose that

A = {[w_{1}, …, w_{k+1}] ∈ CIP^{k} : A(w) = 0}

is a projective hyperplane, where A(w) = a_{1}w_{1} + ⋯ + a_{k+1}w_{k+1}, such that h(C) ⊄ A. The counting function N(h, A) is given by

N(h, A)(r) = N(h, A, r) = ∫_{1}^{r} n(h, A, s) \frac{ds}{s} ,  (2)

where n(h, A, s) equals the number of solutions of the equation A ∘ h̃ = 0 in the disk Δ_{s} = {z ∈ C : |z| < s} counting multiplicities. Jensen’s formula tells us that

N(h, A, r) = \frac{1}{2π} ∫_{0}^{2π} log |A ∘ h̃(re^{iθ})|dθ + N(h, A, 1) ,  r > 1 .  (3)

We hence have the First Main Theorem

N(h, A) ≤ T (h) + O(1) .  (4)

We also consider the truncated counting functions

N_{ℓ}(h, A)(r) = ∫_{1}^{r} n_{ℓ}(h, A, s) \frac{ds}{s}  (ℓ ∈ ZZ^{+}) ,  (5)

where n_{ℓ}(h, A, s) is the number of solutions of A∘h̃ = 0 in Δ_{s} with the multiplicity truncated at ℓ; specifically, if Div A∘h̃|_{Δ_{s}} = ∑ m_{j}{x_{j}}, then n_{ℓ}(h, A, s) = ∑ min{m_{j}, ℓ}. We also write N̄(h, A) = N_{1}(h, A).

For convenience in stating the Second Main Theorem, we use the notation S(h) to denote a function satisfying the estimate

S(h)(r) ≤ O (log T (h, r)) + o(log r)  (6)

for r outside a set of finite Lebesgue measure in IR^{+}.

Our proof of Theorem 1 uses the following Second Main Theorem of H. Cartan [1]:

Cartan’s Theorem. Let h : C → CIP^{k} be a non-degenerate holomorphic curve, and let A_{1}, …, A_{q} be hyperplanes in general position. Then

(q − k − 1)T (h) ≤ ∑_{j=1}^{q} N(h, A_{j}) − N(W_{h}, 0) + S(h)

≤ ∑_{j=1}^{q} N_{k}(h, A_{j}) + S(h) ,

where W_{h} is the Wronskian of a representation h̃ : C → C^{k+1} ^{\} {0} of h.

Here, by h non-degenerate, we mean that h(C) is not contained in a projective hyperplane in CIP^{k}. The hyperplanes A_{j} are said to be in general position if the intersection of any k +1 of them is empty (assuming q > k + 1).

We let \overline{C} = C ∪ \{∞\}, and we identify [w_{1}, w_{2}] ∈ CIP^{1} with w_{2}/w_{1} ∈ \overline{C}. If h : C → \overline{C} is a nonconstant meromorphic function, we write

N(R_{h}) = N(W_{(h_{1},h_{2})}, 0) = N(f′, 0) + 2N(f, ∞) − N(f′, ∞) ,  (7)

where h = h_{2}/h_{1} with h_{1}, h_{2} entire. Cartan’s Theorem for k = 1 becomes the Second Main Theorem of Nevanlinna [7]

\begin{aligned}(q − 2)T (h) & ≤ & ∑_{j=1}^{q} N(h, a_{j}) − N(R_{h}) + S(h) \\ & ≤ & ∑_{j=1}^{q} N̄(h, a_{j}) + S(h) \end{aligned}  (8)

for distinct points a_{1}, …, a_{q} in \overline{C}. In particular, if q = 0, we have

N(R_{h}) ≤ 2T (h) + S(h) .  (9)

We shall use the following elementary fact:

Lemma 2. Let A_{j} : C^{3} → C, j = 1, …, 4, be linear functions such that A_{1}, A_{2}, A_{3} are linearly independent and A_{3}, A_{4} are also linearly independent. Let F : C → CIP^{2} be a nonconstant holomorphic curve, and consider the meromorphic functions

ψ_{1} = (A_{2}/A_{1}) ∘ F, ψ_{2} = (A_{4}/A_{3}) ∘ F .

Then

T (ψ_{1}) ≤ T (F) + O(1) ≤ T (ψ_{1}) + T (ψ_{2}) + O(1) .

Proof: Choose constants c_{1}, …, c_{4} (not all zero) so that

c_{1}A_{1} + c_{2}A_{2} = c_{3}A_{3} + c_{4}A_{4} .

By hypothesis, c_{4} ≠ 0; by interchanging the indices 1 and 2 if necessary, we can also assume that c_{1} ≠ 0. Let A′_{1} = c_{1}A_{1} + c_{2}A_{2}. Since composing a meromorphic function with a fractional linear transformation on \overline{C} only changes its growth function by a bounded term, we may assume instead that

ψ_{1} = (A_{2}/A′_{1}) ∘ F, ψ_{2} = (A_{3}/A′_{1}) ∘ F .

By applying a linear change of variables to CIP^{2} (which changes T (F) only by a bounded term) we can assume that A′_{1} = w_{1}, A_{2} = w_{2}, A_{3} = w_{3}. Let F̃ = (F̃_{1}, F̃_{2}, F̃_{3}) : C → C^{3} − {0} be a regular representation of F. Then ψ_{1} = F̃_{2}/F̃_{1}, ψ_{2} = F̃_{3}/F̃_{1} and we have

T (ψ_{1}, r) ≤ \frac{1}{4π} ∫_{0}^{2π} log (|F̃_{1}(re^{iθ})|^{2} + |F̃_{2}(re^{iθ})|^{2}) dθ
≤ \frac{1}{4π} ∫_{0}^{2π} log ∥F̃(re^{iθ})∥^{2}dθ
= T (F, r) .

To show the second inequality of the lemma, we first note that

log ∥F̃∥^{2} = log(1 + |ψ_{1}|^{2} + |ψ_{2}|^{2}) + log |F̃_{1}|^{2}
≤ log(1 + |ψ_{1}|^{2}) + log(1 + |ψ_{2}|^{2}) + log |F̃_{1}|^{2} + log 2 .

We then integrate over the circle |z| = r to obtain

T (F) ≤ T (ψ_{1}) − N(ψ_{1}, ∞) + T (ψ_{2}) − N(ψ_{2}, ∞) + N(F, w_{1}) + O(1) .

Since a zero of F̃_{1} = w_{1} ∘ F of order k must be a pole of order k of either ψ_{1} or ψ_{2} we have

N(F, w_{1}) ≤ N(ψ_{1}, ∞) + N(ψ_{2}, ∞) ,

which gives the desired inequality. □

3. Proof of the uniqueness theorem.

Our proof of Theorem 1 is an application of Cartan’s Theorem with k = 2, as well as Nevanlinna’s Second Main Theorem (8). Let

P (t) = t^{n} + at^{n−m} + 1 ∈ C[t]

where n > m > 1 and a ∈ C ^{\} {0} is chosen so that P has distinct roots; then the set S = Div P is as given in Theorem 1. Let f, g be meromorphic functions on C and consider the map

h = [h_{1}, h_{2}, h_{3}] : C → CIP^{2}

where

h_{1} = P (f) , h_{2} = P (g) ,
h_{3} = P (f)[P (g) − 1] = P (f)g^{n−m}(g^{m} + a) .

(Note that (h_{1}, h_{2}, h_{3}) is not a regular representation of h, since the h_{j} will have common zeros and may have poles.) We also write

h_{4} = P (g)[P (f) − 1] = h_{1} − h_{2} + h_{3} .

We consider the four projective lines in general position in CIP^{2}:

\begin{matrix}L_{j} = \{[w_{1}, w_{2}, w_{3}] ∈ CIP^{2} : w_{j} = 0\}, j = 1, 2, 3, \\ L_{4} = \{[w_{1}, w_{2}, w_{3}] ∈ CIP^{2} : w_{1} − w_{2} + w_{3} = 0\} . \end{matrix}

Then, by Cartan’s Theorem (with k = 2)

T (h) ≤ ∑_{j=1}^{4} N_{2}(h, L_{j}) + S(h)  (10)

unless h(C) is contained in a projective line. In fact, (10) is valid unless h(C) is contained in one of the three “diagonal” lines:

D_{1} = {w_{1} + w_{3} = 0}, D_{2} = {w_{2} − w_{3} = 0}, D_{3} = {w_{1} − w_{2} = 0} .

(See Figure 1.) If h(C) is contained in any other projective line E, then E intersects ⋃ L_{j} in at least three points, and Nevanlinna’s Second Main Theorem (8) yields a stronger inequality than (10). (The image of h cannot lie in any of the L_{j} since f and g are not constant.)

Figure 1.

Suppose first that f^{−1}(S) = g^{−1}(S) as in part (a). Since h(z_{0}) ∈ L_{1} if and only if either h_{1} has a higher order zero at z_{0} than both h_{2} and h_{3} (which by hypothesis can happen only if f(z_{0}) − x has a multiple zero, for some x ∈ S) or either h_{2} or h_{3} has a higher order pole at z_{0} than h_{1}. Therefore

\begin{aligned}N_{2}(h, L_{1}) & ≤ & N(R_{f}) + 2N̄(g, ∞) \\ & ≤ & 2T (f) + 2T (g) + S(f) , \end{aligned}  (11)

by Nevanlinna’s theorems (4) and (9). Similarly

N_{2}(h, L_{2}) ≤ 2T (f) + 2T (g) + S(g) .  (12)

To estimate N_{2}(L_{3}), we note that if z_{0} is a pole of h_{1} or h_{2}, then it is a pole of h_{3} of order no less than that of h_{1} or of h_{2}. Hence only the excess zeros of h_{3} matter, and we obtain

\begin{aligned}N_{2}(h, L_{3}) & ≤ & N(R_{f}) + N_{2}(0, P (g) − 1) \\ & ≤ & N(R_{f}) + 2N̄(g, 0) + N_{2}(g^{m} + a, 0) \\ & ≤ & 2T (f) + (m + 2)T (g) + S(f) . \end{aligned}  (13)

Similarly

N_{2}(h, L_{4}) ≤ 2T (g) + (m + 2)T (f) + S(g) .  (14)

The last inequality in (13) follows from (4), (9), and the fact that

N_{2}(g^{m} + a, 0) = ∑_{j=0}^{m−1} N_{2}(g, ω^{j}b) ,

where

b^{m} = −a ,  ω = e^{2πi/m} .  (15)

Let us write

T̂ = T (f) + T (g) .  (16)

Combining (11)-(14), we have

∑_{j=1}^{4} N_{2}(h, L_{j}) ≤ (m + 8)T̂ + S(f) + S(g)  (17a)

For part (b), where P (f) and P (g) have the same zeroes with the same multiplicities, we have N(R_{f}) = N(R_{g}) = 0 so that (11)-(14) become:

\begin{aligned}N_{2}(h, L_{1}) & ≤ & 2T (g) \\ N_{2}(h, L_{2}) & ≤ & 2T (f) \\ N_{2}(h, L_{3}) & ≤ & (m + 2)T (g) \\ N_{2}(h, L_{4}) & ≤ & (m + 2)T (f) \end{aligned}

and thus

∑_{j=1}^{4} N_{2}(h, L_{j}) ≤ (m + 4)T̂ .  (17b)

For part (c) we instead set N̄(f, ∞) = N̄(g, ∞) = 0 in the calculations for part (a). Since f and g are entire and thus by Nevanlinna’s Second Main Theorem (8) (with q = 1 and a_{1} = ∞),

N(R_{f}) ≤ T (f) + S(f) , N(R_{g}) ≤ T (g) + S(g) ,

we obtain

N_{2}(h, L_{1}) ≤ T (f) + S(f)
N_{2}(h, L_{2}) ≤ T (g) + S(g)
N_{2}(h, L_{3}) ≤ T (f) + (m + 2)T (g) + S(f)
N_{2}(h, L_{4}) ≤ T (g) + (m + 2)T (f) + S(g) .

Thus

∑_{j=1}^{4} N_{1}(h, L_{j}) ≤ (m + 4)T̂ + S(f) + S(g) .  (17c)

Finally for part (d) we have N_{2}(h, L_{1}) = N_{2}(h, L_{2}) = 0 and N_{2}(h, L_{3}) ≤ (m + 2)T (g), N_{2}(h, L_{4}) ≤ (m + 2)T (f) as in part (b); thus

∑_{j=1}^{4} N_{2}(h, L_{j}) ≤ (m + 2)T̂  (17d)

Recalling that

f^{n} + af^{n−m} = \frac{h_{1} − h_{2} + h_{3}}{h_{2}} , g^{n} + ag^{n−m} = \frac{h_{3}}{h_{1}}

we conclude from the first inequality of Lemma 2, that

nT (f) = T (f^{n} + af^{n−m}) + O(1)
≤ T (h) + O(1) .

Similarly

nT (g) ≤ T (h) + O(1)

and thus

\frac{n}{2}T̂ ≤ T (h) + O(1) .  (18)

Let

Ŝ = S(f) + S(g)  (19)

It follows from (14) and the second inequality of Lemma 2 that

T (h) ≤ T (f^{n} + af^{n−m}) + T (g^{n} + ag^{n−m}) + O(1) = nT̂ + O(1) ,  (20)

and thus

S(h) ≤ Ŝ .  (21)

Combining (10), (18), (20), (21), we obtain

\frac{n}{2}T̂ ≤ ∑_{j=1}^{4} N_{2}(h, L_{j}) + Ŝ .  (22)

Thus by (17a-d),

\frac{n}{2}T̂ ≤ (m + λ)T̂ + Ŝ  (23)

where λ = 8 in part (a), λ = 4 in parts (b) and (c), and λ = 2 in parts (d). Thus

n ≤ 2m + 2λ

which contradicts the hypotheses of Theorem 1.

Hence h(C) must be contained in one of the lines D_{1}, D_{2}, D_{3}. The conclusion now follows by the argument in Li-Yang [4] (with a minor modification since it is assumed in [4] that (n, m) = 1). For completeness, we include the proof below.

Case 1: If h(C) ⊂ D_{1}, then
0 = h_{1} + h_{3} = P (f)P (g) ,

which is impossible.

Case 2: If h(C) ⊂ D_{2}, then h_{2} − h_{3} = 0 and thus

g^{n−m}(g^{m} + a) = P (g) − 1 = \frac{P (g)}{P (f)} .  (24)

Then each zero of g^{n−m}(g^{m} + a) is a pole of P (f) and thus is of order nk for some k ∈ ZZ^{+}. Thus if g − ω^{j}b has a zero of order μ (where b, ω are given by (15) and 0 ≤ j ≤ m − 1), then μ ≥ n. Furthermore, if g has a zero of order ν, then (n − m)ν = nk, and hence n|mν so that ν ≥ \frac{n}{m}. (Actually, ν ≥ n/(n, m).)

Suppose first that m ≥ 2. By Nevanlinna’s Second Main Theorem,

(m − 1)T (g) ≤ N̄(g, 0) + ∑_{j=0}^{m−1} N̄(g, ω^{j}b) + S(g)
≤ \frac{m}{n}N(g, 0) + \frac{1}{n} ∑_{j=0}^{m−1} N(g, ω^{j}b) + S(g)
≤ \frac{2m}{n}T (g) + S(g) .

Therefore m − 1 ≤ \frac{2m}{n}, and thus n ≤ 2m, contradicting our hypotheses.

If m = 1 and g is entire, then

T (g) ≤ N̄(g, 0) + N̄(g, −a) + N̄(g, ∞) + S(g)
≤ \frac{1}{n}N(g, 0) + \frac{1}{n}N(g, −a) + S(g)
≤ \frac{2}{n}T (g) + S(g) ,

and thus n ≤ 2, which contradicts the hypotheses.

Case 3: If h(C) ⊂ D_{3}, then h_{1} − h_{2} = 0 and therefore

f^{n−m}(f^{m} + a) = g^{n−m}(g^{m} + a) .  (25)

Let φ = f/g and suppose that φ is not constant. Then

g^{m}(φ^{n} − 1) = −a(φ^{n−m} − 1) .  (26)

Suppose first that m ≥ 2. Let ξ_{1} …, ξ_{ℓ} be the n-th roots of unity that are not m-th roots of unity. By (26), every zero of φ − ξ_{j} (1 ≤ j ≤ ℓ) must be a pole of g and thus has order at least m. By the Second Main Theorem,

\begin{aligned}(ℓ − 2)T (φ) & ≤ ∑_{j=1}^{ℓ} N̄(φ, ξ_{j}) + S(φ) \\ & ≤ \frac{ℓ}{m}T (φ) + S(φ) , \end{aligned}

which yields ℓ − 2 ≤ \frac{ℓ}{m} and hence ℓ ≤ \frac{2m}{m−1}. Since ℓ ≥ 2(\frac{n}{m} − 1), we obtain

n ≤ m + \frac{m^{2}}{m − 1} ≤ 2m + 2 ,

contradicting our hypotheses. Therefore f/g equals a constant c. It then follows immediately from (25) that c^{n} = c^{m} = 1.

Finally, if m = 1 and f, g are entire, then by (26), φ omits the values e^{2πij/n}, 1 ≤ j ≤ n − 1, which is impossible if n ≥ 4 Hence f/g = c = 1. □

Remark: If we assume instead that m = 1 in part (a) or (b) of Theorem 1, then it follows that P (f) = P (g). To prove this identity, we need to show that Case 2 cannot occur. Suppose on the contrary that h(C) ⊂ D_{2} so that (24) holds. Then every zero of g and of g + a is of order at least n, and the same holds for the zeros of f and of f + a. Also by (24), every pole of f is a zero of either g or g + a; thus

N̄(f, ∞) ≤ N̄(g, 0) + N̄(g, −a) .

Similarly

N̄(g, ∞) ≤ N̄(f, 0) + N̄(f, −a) .

Therefore by (8),

T (g) ≤ N̄(g, 0) + N̄(g, −a) + N̄(g, ∞) + S(g)
≤ N̄(g, 0) + N̄(g, −a) + N̄(f, 0) + N̄(f, −a) + S(g)
≤ \frac{2}{n}T (g) + \frac{2}{n}T (f) + S(g) = \frac{2}{n}T̂ + S(g) .

Similarly T (f) ≤ \frac{2}{n} T̂ + S(f) and hence

T̂ ≤ \frac{4}{n}T̂ + Ŝ ,

which contradicts the assumption that n ≥ 11.

However, we cannot conclude in this case that f = g. For example, consider P (t) = t^{n} + at^{n−1} + 1 with arbitrary n > 1, and let

f = \frac{φ^{n−1} − ψ^{n−1}}{ψ^{n} − φ^{n}} aφ , g = \frac{φ^{n−1} − ψ^{n−1}}{ψ^{n} − φ^{n}} aψ ,

where φ, ψ are arbitrary distinct meromorphic functions. Then P (f) = P (g), but f ≠ g.

4. Uniqueness Polynomials.

A polynomial P ∈ C[t] is called a uniqueness polynomial for meromorphic functions if

P (f) = P (g) ⇒ f = g

for all nonconstant meromorphic functions f, g on C. For example, we demonstrated in the last part (Case 3) of the proof of Theorem 1 that

t^{n} + at^{n−m}

is a uniqueness polynomial for meromorphic functions (the constant term is immaterial) if n ≥ 2m + 3 and (n, m) = 1. The following theorem says that generic polynomials of degree at least 5 are uniqueness polynomials for meromorphic functions.

We say that a statement S(P) holds for a generic polynomial P ∈ C[t] of degree n if there exists a nonzero polynomial Φ ∈ C[X_{0}, …, X_{n}] such that S(P) holds for

P (t) = u_{n}t^{n} + ⋯ + u_{1}t + u_{0} .

whenever Φ(u_{0}, …, u_{n}) ≠ 0.

Theorem 3. If P is a generic polynomial of degree n ≥ 5, then

P (f) = P (g) ⇒ f = g  (∗)

for all nonconstant meromorphic functions f, g on C. Furthermore if P is a generic polynomial of degree 4, then (∗) holds for all nonconstant entire functions f, g.

Proof: We can consider without loss of generality monic polynomials without constant terms. Suppose

P (t) = t^{n} + u_{n−1}t^{n−1} + ⋯ + u_{1}t .

We consider the affine algebraic curve

C_{P} = \{(z, w) ∈ C^{2} : \frac{P (z) − P (w)}{z − w} = 0\}  (27)

and its closure in CIP^{2},

C̄_{P} = \{[v, z, w] ∈ CIP^{2} : \frac{v^{n}[P (z/v) − P (w/v)]}{z − w} = 0\} .  (28)

We first observe that it suffices to show that C̄_{P} is smooth (i.e., is a 1-dimensional complex submanifold) for generic P. Suppose we are given a polynomial P of degree ≥ 5 so that C̄_{P} is smooth. Suppose f, g are meromorphic on C with P (f) = P (g). Consider the map

F = [1, f, g] : C → CIP^{2} .

If f ≢ g, then F (C) ⊂ C̄_{P}. Since C̄_{P} is smooth and deg C̄_{P} = deg P − 1 ≥ 4, it follows that C̄_{P} is a Riemann surface of genus ≥ 3. Thus F : C → C̄_{P} is constant, and therefore f and g are constant. If deg P = 4 and f, g are entire, then genus C̄_{P} = 1 and F : C → C_{P} is constant. Therefore the theorem holds for all polynomials P with C̄_{P} smooth.

We now show that C̄_{P} is smooth for generic P. Let H_{∞} = {[v, z, w] ∈ CIP^{2} : v = 0}. Then

C̄_{P} ∩ H_{∞} = \{[0, v, w] : \frac{z^{n} − w^{n}}{z − w} = 0\} = \{[0, 1, ζ^{j}] : 1 ≤ j ≤ n − 1\} ,

where ζ = e^{2πi/n}. We note that the points at infinity (0, 1, ζ^{j}) (1 ≤ j ≤ n − 1) are smooth points of C̄_{P}, since

\frac{∂}{∂z} (\frac{v^{n}[P (z/v) − P (w/v)]}{z − w})|_{(0,1,ζ^{j})} = n(1 − ζ^{j})^{−2} ≠ 0

for 1 ≤ j ≤ n − 1. (Alternately, one can use Bezout’s Theorem to conclude that C̄_{P} is smooth at the n − 1 points of C̄_{P} ∩ H_{∞}.) Therefore, it suffices to show that the affine curve C_{P} is smooth for generic P.

We identify the point (u_{1}, …, u_{n−1}) ∈ C^{n−1} with the polynomial

P (t) = t^{n} + u_{n−1}t^{n−1} + ⋯ + u_{1}t ∈ C[t] ,

and we consider the (polynomial) function F : C^{n−1} × C^{2} → C given by

F (P, z, w) = \frac{P (z) − P (w)}{z − w} = u_{1} + u_{2}(z + w) + u_{3}(z^{2} + zw + w^{2}) + · · · .

Then F^{−1}(0) is a smooth complex n-dimensional manifold. Let

π : F^{−1}(0) → C^{n−1}

be the projection. Then π is surjective and π^{−1}(P) = C_{P} for all P ∈ C^{N−1}. Therefore, for generic P, the fiber C_{P} is smooth. □

Remark: Suppose that P (t) = t^{n} + at^{n−m} + 1 as in Theorem 1 and assume that (n, m) = 1. An alternate demonstration that P (f) = P (g) ⇒ f = g (Case 3 of the proof of Theorem 1) goes as follows: The complex curve C̄_{P} given by (28) can be shown to be irreducible (using the assumption that (n, m) = 1), but it is not smooth. However, by the proof of Theorem 3 it suffices to show that the genus g_{C} of the normalization C̃ →^{ρ} C̄_{P} is greater than 1. Applying the Riemann-Hurwitz formula (see [2, p. 219]) to π ∘ ρ : C̃ → \overline{C} where π([v, z, w]) = w/z, we conclude that

g_{C} = (m − 1) (n − \frac{m}{2} − 2) ,  (29)

and thus g_{C} ≥ 2 if n ≥ m + 3, m ≥ 2. (Note that C̄_{P} has only one ordinary singularity at [1, 0, 0] of multiplicity n − m − 2. Equation (29) can alternately be obtained by blowing up [1, 0, 0] and applying the adjunction formula (see [2, p. 147]).) If f, g are entire and n ≥ 4, the genus is irrelevant and the irreducibility of C_{P} is sufficient, since the map F constructed above omits n − 1 points at infinity.

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Department of Mathematics Johns Hopkins University Baltimore, MD 21218 Electronic mail: shiffman@math.jhu.edu