SQUAREFREE VALUES OF MULTIVARIABLE POLYNOMIALS

BJORN POONEN

Abstract. Given f ∈ Z[x_{1}, …, x_{n}], we compute the density of x ∈ Z^{n} such that f(x) is squarefree, assuming the abc conjecture. Given f, g ∈ Z[x_{1}, …, x_{n}], we compute unconditionally the density of x ∈ Z^{n} such that gcd(f(x), g(x)) = 1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f ∈ Z[x], we estimate the size of the image of f({1, 2, …, n}) in (Q^{∗}/Q^{∗2}) ∪ {0}.

1. Introduction

An integer n is called squarefree if for all prime numbers p we have p^{2} ∤ n (that is, p^{2} does not divide n). Heuristically, one expects that if one chooses a positive integer n “at random,” then for each prime p, the “probability” that p^{2} ∤ n equals 1 − p^{−2}; and the assumption that these probabilities are “independent” leads to the guess that the density of squarefree positive integers equals

∏_{prime p}(1 − p^{−2}) = ζ(2)^{−1} = 6/π^{2},

where ζ(s) is the Riemann zeta function, defined by

ζ(s) = ∑_{n≥1} n^{−s} = ∏_{prime p}(1 − p^{−s})^{−1}

for Re s > 1. One can formulate this guess precisely by defining the density of a set of positive integers S as

μ(S) := lim_{B→∞} \frac{#(S ∩ [1, B])}{B}.

In fact, the guess can be proved by simple sieve techniques [HW79, §18.6].

Now suppose that f(x) is a polynomial with integer coefficients, and let S be the set of positive integers n for which f(n) is squarefree. This time one guesses that the density of S equals ∏_{prime p}(1 − c_{p}/p^{2}) where c_{p} equals the number of integers n ∈ [0, p^{2} − 1] for which p^{2} | f(n). When deg f ≤ 2, a simple sieve again shows that the guess is correct. When deg f = 3, a more complicated argument is needed (see [Hoo67], or, for an improved error term, Chapter 4 of [Hoo76]). For general f with deg f ≥ 4, it is unknown whether the heuristic conjecture is correct, but A. Granville [Gra98] showed that it follows from the abc conjecture. (Recall that the abc conjecture is the statement that for any ϵ > 0, there exists a constant C = C(ϵ) > 0 such that if a, b, c are coprime positive integers satisfying a+b = c, then c < C( ∏_{p|abc} p)^{1+ϵ}.) Granville used the abc conjecture in conjunction with Belyi’s

Date: September 28, 2003 (revised slightly after publication).

2000 Mathematics Subject Classification. Primary 11C08.

This research was supported by NSF grant DMS-9801104 and a Packard Fellowship. Part of the research was done while the author was enjoying the hospitality of the Université de Paris-Sud. This article has been published in Duke Math. J. 118 (2003), no. 2, 353–373.

Theorem, to bound the number of polynomial values divisible by the square of a large prime. He also proved a conditional result for homogeneous polynomials in two variables, extending some earlier results along these lines, such as [Gre92]. (See [Gra98] for more references; some of these earlier results were unconditional in low degree cases.)

In this paper we generalize Granville’s results to arbitrary polynomials over Z in many variables, still assuming the abc conjecture. The proof proceeds by reduction to the onevariable case, and the abc conjecture is required only because it is used by Granville; it is not required for the reduction. Such a fibering argument was used also in [GM91]. One defect of our proof is that it appears not to work for the most natural generalization of density in the multivariable case: see Section 2 and the remark following the proof of Lemma 6.2 for more details. An application of our result is towards estimating, given a regular quasiprojective scheme X over Z, what fraction of hypersurface sections of X are regular. (See [Poo02].)

If F_{q} is a finite field of characteristic p, we prove an analogue for polynomials over F_{q}[t] unconditionally, using a completely different proof, exploiting the fact that F_{q}[t] has an F_{q}[t^{p}]-linear derivation. One application of this result, suggested by A. J. de Jong [dJ02, §4.22], is to counting elliptic curves with squarefree discriminant: see Section 3. The case of squarefree values of a separable irreducible one-variable polynomial over F_{q}[t] (or more generally k^{th}-power-free values for polynomials over the ring of regular functions on any affine curve over F_{q}) was proved earlier by K. Ramsay [Ram92]^{1} using a lemma of N. Elkies involving a derivation. In Section 8, we sketch a generalization of our result to multivariable polynomials over such rings of regular functions.

A related problem asks, given relatively prime polynomials f(x_{1}, …, x_{n}) and g(x_{1}, …, x_{n}) over Z, what is the density of n-tuples of positive integers for which the values of f and g are relatively prime? Again there is a heuristic guess, and it was proved in [Eke91] that this guess is correct. We generalize by using a stronger definition of density (involving boxes of arbitrary dimensions, instead of only equal dimensions as considered in [Eke91]) and by simultaneously proving the function field analogue. The generalizations are needed to prove the corresponding results about squarefree values.

Finally, we confirm a guess made in [Gra98], namely that for a nonzero polynomial f(x) ∈ Z[x], the size of the image of {f(1), f(2), …, f(B)} in (Q^{∗}/Q^{∗2}) ∪ {0} is c_{f}B + o(B) as B → ∞, for some constant c_{f} depending on f. Moreover, we find an explicit formula for c_{f}. In particular, c_{f} = 1 if f is squarefree of degree ≥ 2.

2. Definition of density

In Sections 2 through 7, A denotes Z or F_{q}[t] for some prime power q = p^{e}. Let K denote the fraction field of A. For nonzero a ∈ A define |a| := #(A/a), and define |0| = 0. If 𝔭 is a nonzero prime of A, let |𝔭| := #(A/𝔭). Define

Box = Box(B_{1}, …, B_{n}) = \begin{cases}\{ (a_{1}, …, a_{n}) ∈ Z^{n} : 0 < a_{i} ≤ B_{i} for all i \} & if A = Z, \\ \{ (a_{1}, …, a_{n}) ∈ A^{n} : |a_{i}| ≤ B_{i} for all i \} & if A = F_{q}[t]. \end{cases}

1The formula for the density in Theorem 1 of [Ram92] should read Z = ∏_{v∉S} (1 − ρ(kv)/∥v∥^{k}). The proof there is correct, but the statement is unfortunately misprinted, with ρ(v) in place of ρ(kv).

For S ⊆ A^{n}, define

μ(S) := lim_{B_{1},...,B_{n}→∞} \frac{#(S ∩ Box)}{#Box},

and define \overline{μ}(S) and \underline{μ}(S) similarly using lim sup and lim inf in place of lim. If a subset S ⊆ Z^{n} and its 2^{n} reflections in the coordinate hyperplanes have a common density in this strong sense, then we can estimate #(S ∩ R)/#R for regions R of many other shapes. For instance, if R_{B} is the ball of radius B centered at the origin, then #(S ∩ R_{B})/#R_{B} → μ(S) as B → ∞, since R_{B} can be approximated by a Boolean combination of k boxes and their reflections, with an error of at most ϵ_{k}B^{n} lattice points for B large relative to k, where ϵ_{k} → 0 as k → ∞.

In some of our results we can prove that the density exists only in a weaker sense. Define

\overline{μ}_{n}(S) := lim sup_{B_{1},...,B_{n−1}→∞} lim sup_{B_{n}→∞} \frac{#(S ∩ Box)}{#Box}.

This has the effect of considering only boxes in which the n^{th} dimension is large relative to the others. Define \underline{μ}_{n}(S) similarly. If \overline{μ}_{n}(S) = \underline{μ}_{n}(S), define μ_{n}(S) as the common value. Also define

\overline{μ}_{weak}(S) := max_{σ} lim sup_{B_{σ(1)}→∞} ⋯ lim sup_{B_{σ(n)}→∞} \frac{#(S ∩ Box)}{#Box},

where σ ranges over permutations of {1, 2, …, n}. This definition in effect considers only boxes whose dimensions can be ordered so that each is very large relative to the previous dimensions. Define \underline{μ}_{weak}(S) similarly, and define μ_{weak}(S) if \overline{μ}_{weak}(S) = \underline{μ}_{weak}(S).

3. Theorems

Throughout this paper, p represents a prime number. In particular, in a sum or product indexed by p, it is assumed that p runs through only primes. Similarly, 𝔭 represents a nonzero prime of A.

Theorem 3.1 (Relatively prime values). Let f, g ∈ A[x_{1}, …, x_{n}] be polynomials that are relatively prime as elements of K[x_{1}, …, x_{n}]. Let

R_{f,g} := { a ∈ A^{n} : gcd(f(a), g(a)) = 1 }.

Then μ(R_{f,g}) = ∏_{𝔭}(1 − c_{𝔭}/|𝔭|^{n}), where 𝔭 ranges over all nonzero primes of A, and c_{𝔭} is the number of x ∈ (A/𝔭)^{n} satisfying f(x) = g(x) = 0 in A/𝔭.

The assumptions and conclusions for the squarefree value theorem differ slightly in the Z and F_{q}[t] cases, so we separate them into Theorem 3.2 and Theorem 3.4.

Theorem 3.2 (Squarefree values over Z). Assume the abc conjecture. Let f ∈ Z[x_{1}, …, x_{n}] be a polynomial that is squarefree as an element of Q[x_{1}, …, x_{n}], and suppose that x_{n} appears in each irreducible factor of f. Let

S_{f} := { a ∈ Z^{n} : f(a) is squarefree }.

For each prime p, let c_{p} be the number of x ∈ (Z/p^{2})^{n} satisfying f(x) = 0 in Z/p^{2}. Then μ_{n}(S_{f}) = ∏_{p}(1 − c_{p}/p^{2n}).

If the degree of x_{n} in each irreducible factor of f in Theorem 3.2 is ≤ 3, then it is unnecessary to assume the abc conjecture, because the proof reduces to the case of onevariable polynomials of degree ≤ 3, for which an unconditional result is known [Hoo67].

The n = 1 case of Theorem 3.2 differs slightly from Theorem 1 in [Gra98] in that the latter computes the density of squarefree values of f(x)/m where m is a particular positive integer dividing all values of f. Such results can be proved in the multivariable case just as easily as Theorem 3.2; the key to all such results is Lemma 6.2.

Corollary 3.3. Let notation and assumptions be as in Theorem 3.2 but without the restriction that x_{n} appears in f. Then μ_{weak}(S_{f}) = ∏_{p}(1 − c_{p}/p^{2n}).

Corollary 3.3 for f(x_{1}, …, x_{n}) follows from Theorem 3.2 applied to f(x_{σ(1)}, …, x_{σ(n)}) for all permutations σ, since in the definition of μ_{weak} we may discard each lim sup corresponding to a variable that does not appear.

Theorem 3.4 (Squarefree values over F_{q}[t]). Let A = F_{q}[t]. Let f ∈ A[x_{1}, …, x_{n}] be a polynomial that is squarefree as an element of K[x_{1}, …, x_{n}]. Let

S_{f} := { a ∈ A^{n} : f(a) is squarefree }.

For each nonzero prime 𝔭 ⊆ A, let c_{𝔭} be the number of x ∈ (A/𝔭^{2})^{n} satisfying f(x) = 0 in A/𝔭^{2}. Then μ(S_{f}) = ∏_{𝔭}(1 − c_{𝔭}/|𝔭|^{2n}).

Remark. Note in particular that Theorem 3.4 proves a result for μ instead of only for μ_{n}.

Suppose gcd(q, 6) = 1. One application of Theorem 3.4 is to computing asymptotics for the weighted number R_{d} of isomorphism classes of elliptic curves (E, O) over F_{q}[t] with squarefree discriminant, as d → ∞ for fixed q [dJ02, §4.22]. “Weighted” means that each isomorphism class receives the weight 1/# Aut(E, O) instead of 1. This number R_{d} is closely connected to the density of (A, B) ∈ F_{q}[t]^{2} such that the discriminant Δ = −16(4A^{3} +27B^{2}) of y^{2} = x^{3} + Ax + B is squarefree, except that one works with homogeneous polynomials A ∈ H^{0}(P^{1}, O(4d)) and B ∈ H^{0}(P^{1}, O(6d)), so that the density has a factor corresponding to the point at infinity on P^{1} in addition to the affine points. A calculation shows that the density of such (A, B) having a double zero at a particular closed point 𝔭 of P^{1} = P_{F_{q}}^{1} is (2|𝔭|^{2} − |𝔭|)/|𝔭|^{4}, where |𝔭| denotes the size of the residue field of 𝔭; from this and our methods we obtain

lim_{d→∞} \frac{R_{d}}{q^{10d+1}} = \frac{q}{q − 1} ∏_{𝔭∈P^{1}} (1 − \frac{2|𝔭|^{2} − |𝔭|}{|𝔭|^{4}}) ,

and the number γ_{q} of [dJ02, §4.22] equals

γ_{q} = \frac{q^{3}}{(q − 1)^{2}(q + 1)} ∏_{𝔭∈P^{1}} (1 − \frac{2|𝔭|^{2} − |𝔭|}{|𝔭|^{4}}) .

Remark. Since −16(4A^{3} + 27B^{2}) has degree only 2 in B, it is possible to obtain the result of the previous paragraph by arguments simpler than those needed for the proof of Theorem 3.4 in general. (This was pointed out to me by de Jong.)

For a generalization of Theorem 3.4 to other rings of functions, see Section 8. Analogues where “squarefree” is replaced by “k^{th}-power-free” follow immediately from the same arguments once one has Lemma 6.2 (or the corresponding function field result).

Theorem 3.5 (Values in Q^{∗}/Q^{∗2}). Let f(x) ∈ Z[x] be a nonzero polynomial. Write f(x) = cg(x)^{2}h(x) where c ∈ Z, g(x) ∈ Z[x], and h(x) is a squarefree polynomial in Z[x] whose coefficients have gcd 1. If deg h > 3, assume the abc conjecture. Then the image of {f(1), f(2), …, f(B)} in (Q^{∗}/Q^{∗2}) ∪ {0} has size c_{f}B + o(B) for some constant c_{f} ∈ [0, 1]. If deg h = 0, then c_{f} = 0. If deg h ≥ 2, then c_{f} = 1. If deg h = 1, say h(x) = ax + b, then

c_{f} = \frac{6}{π^{2}} (∑_{r=0}^{|a|−1} δ_{r}) ∏_{p|a}(1 − p^{−2})^{−1} ∈ \frac{1}{π^{2}} Q,

where δ_{r} := 1/m^{2} if m is the smallest positive integer satisfying m^{2}r ≡ b (mod a), or δ_{r} := 0 if no such m exists.

Assuming the abc conjecture, Granville [Gra98, Corollary 2] proved that the size of the image in Theorem 3.5 was at least some positive constant times B (when f(x) has no repeated roots), and guessed that the size should be asymptotic to a constant times B, as our Theorem 3.5 shows. An essentially equivalent version of Theorem 3.5 has been independently proved by P. Cutter, A. Granville, and T. Tucker (Theorems 1A, 1B, and 1C of [CGT03]), using a similar proof. They also prove a few related results not considered here.

It is natural to ask, as Granville has also done, what the multivariable analogue of Theorem 3.5 should be. Here we formulate a precise question along these lines:

Question 3.6. Suppose f ∈ Z[x_{1}, …, x_{n}] is nonconstant and squarefree as an element of Q[x_{1}, …, x_{n}]. For B ≥ 1, let S_{B} = f({1, 2, …, B}^{n}) ⊂ Z, and let T_{B} be the image of S_{B} in (Q^{∗}/Q^{∗2}) ∪ {0}. Does #T_{B}/#S_{B} tend to a positive limit as B → ∞?

We do not have enough evidence to conjecture an answer. But even if the answer is yes, it is not clear that we would understand the asymptotic size of T_{B}, because even the problem of estimating #S_{B} seems very difficult.

4. Zero values

The following lemma is well-known. We include a proof mainly because it is a toy version of some of the reductions used later on.

Lemma 4.1. Let f ∈ A[x_{1}, …, x_{n}] be a nonzero polynomial. Let Z = { a ∈ A^{n} : f(a) = 0 }. Then μ(Z) = 0.

Proof. We use induction on n. The base case n = 0 is trivial, so suppose n ≥ 1. Let f_{1} ∈ A[x_{1}, …, x_{n−1}] be the leading coefficient of f when f is viewed as a polynomial in x_{n}. Let δ be the x_{n}-degree of f. Now Z ⊆ Z_{1} ∪ Z_{2} where

Z_{1} := { a ∈ A^{n} : f_{1}(a) = 0 },
Z_{2} := { a ∈ A^{n} : f_{1}(a) ≠ 0 and f(a) = 0 }.

By the inductive hypothesis, μ(Z_{1}) = 0. For each (a_{1}, …, a_{n−1}) ∈ A^{n−1}, there are at most δ values a_{n} ∈ A for which (a_{1}, …, a_{n−1}, a_{n}) ∈ Z_{2}. Thus μ(Z_{2}) = 0, by definition of μ. Hence μ(Z) = 0, as desired. □

5. Relatively prime values

The bulk of the work in proving Theorem 3.1 is in the following.

Lemma 5.1. Let f, g ∈ A[x_{1}, …, x_{n}] be polynomials that are relatively prime as elements of K[x_{1}, …, x_{n}]. Let

Q_{f,g,M} := { a ∈ A^{n} : ∃𝔭 such that |𝔭| ≥ M and 𝔭 | f(a), g(a) }.

Then lim_{M→∞} \overline{μ}(Q_{f,g,M}) = 0.

Proof. Since we are interested only in 𝔭 with |𝔭| large, we may divide f and g by any factors in A that they have, in order to assume that f and g are relatively prime as elements of A[x_{1}, …, x_{n}].

The proof will be by induction on n. The case n = 0 is trivial, so assume n ≥ 1. We need to bound the size of Q := Q_{f,g,M} ∩Box, whenever the “dimensions” B_{i} of Box are sufficiently large. Without loss of generality, M ≤ B_{1} ≤ B_{2} ≤ ⋯ ≤ B_{n}. Set B_{0} = M and B_{n+1} = ∞. Let f_{1}, g_{1} ∈ A[x_{1}, …, x_{n−1}] be the leading coefficients of f and g when f and g are viewed as polynomials in x_{n}.

Case 1. One of the polynomials, say g, is a polynomial in x_{1}, …, x_{n−1} only.

In this case, we use an inner induction on δ, where δ is the x_{n}-degree of f. The base case δ = 0 is handled by the outer inductive hypothesis, so from now on assume δ > 0. We may reduce to the case that f and g are irreducible. If g | f_{1}, then we can subtract a multiple of g from f to lower its x_{n}-degree δ, without changing Q_{f,g,M} or the relative primality of f and g, so the result follows from the inner inductive hypothesis. Hence we may assume g ∤ f_{1}. Since g is irreducible, f_{1} and g are relatively prime in A[x_{1}, …, x_{n−1}].

Now Q = ⋃_{s=0}^{n} Q_{s}, where

Q_{s} := { a ∈ Box : ∃𝔭 such that B_{s} ≤ |𝔭| < B_{s+1} and 𝔭 | f(a), g(a) },

so it suffices to show that given 0 ≤ s ≤ n, the ratio #Q_{s}/#Box can be made arbitrarily small by choosing the B_{i} sufficiently large.

Suppose we fix s with 0 ≤ s < n. (We will bound Q_{n} later.) Let X be the subscheme of A_{A}^{n} defined by f = g = 0. Since f and g are relatively prime, X has codimension at least 2 in A_{A}^{n}. Let π : A_{A}^{n} → A_{A}^{s} be the projection onto the first s coordinates. Let Y_{i} be the (constructible) set of y ∈ A_{A}^{s} such that the fiber X_{y} := X ∩ π^{−1}(y) has codimension i in π^{−1}(y) ≃ A_{κ(y)}^{n−s}. (Here κ(y) denotes the residue field of y.) Since X has codimension at least 2 in A_{A}^{n}, it follows from Theorem 15.1(i) of [Mat89] that the subset Y_{i} has codimension at least 2 − i in A_{A}^{s}. In particular, we can choose a nonzero h ∈ A[x_{1}, …, x_{s}] vanishing on Y_{1}. Also we can find relatively prime j_{1}, j_{2} ∈ A[x_{1}, …, x_{s}] vanishing on Y_{0} as follows: choose any nonzero j_{1} vanishing on Y_{0}; if I(Y_{0}) is not contained in the union of the minimal primes over (j_{1}), then any j_{2} ∈ I(Y_{0}) outside those primes will be relatively prime to j_{1}; if I(Y_{0}) is contained in that union, then Proposition 1.11(i) of [AM69] implies that I(Y_{0}) is contained in some minimal prime over (j_{1}), but such a prime has codimension 1, contradicting the fact that Y_{0} has codimension at least 2. Define Y_{≥2} := ⋃_{i≥2} Y_{i}.

Given a = (a_{1}, …, a_{n}) ∈ A^{n} and a nonzero prime 𝔭 of A, let a_{𝔭} = (a_{1}, …, a_{n})_{𝔭} denote the closed point in A_{A/𝔭}^{n} whose coordinates are a_{1}, …, a_{n}. Thus

Q_{s} = { a ∈ Box : ∃𝔭 such that B_{s} ≤ |𝔭| < B_{s+1} and a_{𝔭} ∈ X }.

Let Z := { a ∈ Box : h(a_{1}, …, a_{s}) = 0 }. Define

R_{≥2} := { a ∈ Box : ∃𝔭 such that B_{s} ≤ |𝔭| < B_{s+1}, a_{𝔭} ∈ X, and (a_{1}, …, a_{s})_{𝔭} ∈ Y_{≥2} },

and define R_{1} and R_{0} similarly, using Y_{1} and Y_{0}, respectively, in place of Y_{≥2}.

Then Q_{s} ⊆ Z ∪ R_{≥2} ∪ (R_{1} − Z) ∪ R_{0}. By Lemma 4.1, #Z/#Box can be made arbitrarily small by choosing the B_{i} sufficiently large.

Next consider R_{≥2}. It suffices to show that for (a_{1}, …, a_{s}) ∈ Box(B_{1}, …, B_{s}), the fraction of (a_{s+1}, …, a_{n}) in Box(B_{s+1}, …, B_{n}) for which there exists a prime 𝔭 with B_{s} ≤ |𝔭| < B_{s+1}, a_{𝔭} ∈ X, and (a_{1}, …, a_{s})_{𝔭} ∈ Y_{≥2} is small when B_{s} is large. Fix (a_{1}, …, a_{s}) ∈ Box(B_{1}, …, B_{s}). If 𝔭 is a prime with B_{s} ≤ |𝔭| < B_{s+1} and y := (a_{1}, …, a_{s})_{𝔭} lies in Y_{≥2}, then X_{y} has codimension at least 2 in A_{A/𝔭}^{n−s}, so

#X_{y}(A/𝔭) = O(|𝔭|^{n−s−2}) = O((#(A/𝔭)^{n−s})/|𝔭|^{2}).

Moreover, the implied constant can be made uniform in y, since the X_{y} are fibers in an algebraic family. Since |𝔭| < B_{s+1}, the reductions modulo 𝔭 of the (a_{s+1}, …, a_{n}) ∈ Box(B_{s+1}, …, B_{n}) are almost uniformly distributed in (A/𝔭)^{n−s}: to be precise, each residue class in (A/𝔭)^{n−s} is represented by a fraction at most O(#(A/𝔭)^{−(n−s)}) of these (a_{s+1}, …, a_{n}), where the implied constant depends only on n. Hence the fraction of (a_{s+1}, …, a_{n}) ∈ Box(B_{s+1}, …, B_{n}) satisfying (a_{1}, …, a_{n})_{𝔭} ∈ X_{y} is O(1/|𝔭|^{2}), and summing over all 𝔭 with B_{s} ≤ |𝔭| < B_{s+1} still yields a fraction that can be made arbitrarily small by taking B_{s} large, since ∑_{𝔭} 1/|𝔭|^{2} converges.

We now adapt the previous paragraph to bound #(R_{1} − Z). Suppose (a_{1}, …, a_{s}) ∈ Box(B_{1}, …, B_{s}) and h(a_{1}, …, a_{s}) ≠ 0. Let η be the total degree of h. Then |h(a_{1}, …, a_{s})| = O(B_{s}^{η}), where the constant implied by the O depends only on h, not on the a_{i} or B_{i}. Thus, provided that B_{s} is large, h(a_{1}, …, a_{s}) can be divisible by at most η primes 𝔭 satisfying B_{s} ≤ |𝔭| < B_{s+1}. Hence (a_{1}, …, a_{s})_{𝔭} ∈ Y_{1} for at most η primes 𝔭 satisfying B_{s} ≤ |𝔭| < B_{s+1}. By definition of Y_{1}, if y = (a_{1}, …, a_{s})_{𝔭} for such 𝔭, then X_{y} has codimension at least 1 in A_{A/𝔭}^{n−s}, so #X_{y}(A/𝔭) = O((#(A/𝔭)^{n−s})/|𝔭|), where the implied constant is independent of y. The reductions modulo 𝔭 of the (a_{s+1}, …, a_{n}) ∈ Box(B_{s+1}, …, B_{n}) are again almost uniformly distributed in (A/𝔭)^{n−s}. Hence the fraction of (a_{s+1}, …, a_{n}) in Box(B_{s+1}, …, B_{n}) whose reduction modulo 𝔭 lies in X_{y} is O(1/|𝔭|). Summing over at most η possible primes 𝔭 with |𝔭| ≥ B_{s} still yields a fraction that can be made arbitrarily small by taking B_{s} large.

Finally we consider R_{0}. Since s < n, the outer inductive hypothesis applied to j_{1} and j_{2} implies that #R_{0}/#Box can be made arbitrarily small by taking the B_{i} large.

To finish Case 1, we need to bound Q_{n}. We have Q_{n} ⊆ S_{0} ∪ S ∪ S′, where

S_{0} := { a ∈ Box : g(a_{1}, …, a_{n−1}) = 0 },
S := { a ∈ Box : ∃𝔭 such that |𝔭| ≥ B_{n} and 𝔭 | f_{1}(a), g(a)}
S′ := { a ∈ Box : g(a_{1}, …, a_{n−1}) ≠ 0 and ∃𝔭 such that |𝔭| ≥ B_{n}, 𝔭 | f(a), g(a) and 𝔭 ∤ f_{1}(a) }.

Lemma 4.1 bounds #S_{0}/#Box. The outer inductive hypothesis applied to f_{1} and g bounds #S/#Box.

It remains to bound #S′. For (a_{1}, …, a_{n−1}) ∈ Box(B_{1}, …, B_{n−1}) such that g(a_{1}, …, a_{n−1}) ≠ 0, we will show that the fraction of a_{n} ∈ Box(B_{n}) such that there exists 𝔭 with |𝔭| ≥ B_{n}, 𝔭 | f(a), g(a) and 𝔭 ∤ f_{1}(a) is small. We use a method similar to that used to bound R_{1}. Let γ denote the total degree of g. If B_{n} is sufficiently large (depending only on g), then given (a_{1}, …, a_{n−1}), there are at most γ primes 𝔭 dividing g(a) with |𝔭| ≥ B_{n}. For each such 𝔭, if moreover 𝔭 ∤ f_{1}(a), then the polynomial f(a_{1}, …, a_{n−1}, x_{n}) mod 𝔭 in (A/𝔭)[x_{n}] is of degree δ, and has at most δ roots in A/𝔭. For each such root, there are at most O(1) elements of Box(B_{n}) reducing to it modulo 𝔭, since |𝔭| ≥ B_{n}. Thus given (a_{1}, …, a_{n−1}), there are at most γδ · O(1) = O(1) values of a_{n} ∈ Box(B_{n}) for which (a_{1}, …, a_{n}) ∈ S′. Thus #S′/#Box can be made small by choosing B_{n} large.

Case 2. The x_{n}-degree of f and g are both positive.

Let R ∈ A[x_{1}, …, x_{n−1}] be the resultant of f and g with respect to x_{n}. Since f and g are relatively prime, R is nonzero. Since f_{1}, g_{1}, and R are all nonzero and do not involve x_{n}, none of them are multiples of f or g. Since f and g are irreducible, each of f_{1}, g_{1}, and R must be relatively prime to each of f and g. Moreover, if 𝔭 is a prime dividing f(a) and g(a), and if the leading coefficients f_{1}(a) and g_{1}(a) are nonzero modulo 𝔭, then by a well known property of the resultant, 𝔭 | R(a). Hence

{ a ∈ Box : 𝔭 | f(a), g(a) } ⊆ { a ∈ Box : 𝔭 | f_{1}(a), g(a) }
∪ { a ∈ Box : 𝔭 | f(a), g_{1}(a) }
∪ { a ∈ Box : 𝔭 | f(a), R(a) }.

Taking the union over all 𝔭 with |𝔭| ≥ M and applying Case 1 to f_{1}, g, to f, g_{1}, and to f, R completes the proof. □

Proof of Theorem 3.1. Let P_{M} denote the set of nonzero primes 𝔭 of A such that |𝔭| < M. Approximate R_{f,g} by

R_{f,g,M} := { a ∈ A^{n} : f(a) and g(a) are not both divisible by any prime 𝔭 ∈ P_{M} }.

Define the ideal I as the product of all 𝔭 in P_{M}. Then R_{f,g,M} is a union of cosets of the subgroup I^{n} ⊂ A^{n}. (Here I^{n} is the cartesian product.) Hence μ(R_{f,g,M}) is the fraction of residue classes in (A/I)^{n} in which for all 𝔭 ∈ P_{M}, at least one of f(a) and g(a) is nonzero modulo 𝔭. Applying the Chinese Remainder Theorem shows that μ(R_{f,g,M}) = ∏_{𝔭∈P_{M}}(1 − c_{𝔭}/|𝔭|^{n}). By Lemma 5.1,

μ(R_{f,g}) = lim_{M→∞} μ(R_{f,g,M}) = ∏_{𝔭}(1 − c_{𝔭}/|𝔭|^{n}).

Since f and g are relatively prime as elements of K[x_{1}, …, x_{n}], there exists a nonzero u ∈ A such that f = g = 0 defines a subscheme of A_{A[1/u]}^{n} of codimension at least 2. Thus c_{𝔭} = O(|𝔭|^{n−2}) as |𝔭| → ∞, and the product converges. □

6. Squarefree values of polynomials over Z

If f ∈ A[x_{1}, …, x_{n}], and M ≥ 1, define

T_{f,M} := { a ∈ A^{n} : ∃𝔭 with |𝔭| ≥ M such that 𝔭^{2} | f(a) }.

For the rest of this section, we take A = Z. The following is a variant of Theorem 1 of [Gra98], and has the same proof.

Lemma 6.1. Assume the abc conjecture. Suppose that f ∈ Z[x] is squarefree as a polynomial in Q[x]. For each prime p ≥ M, let c_{p} be the number of x ∈ Z/p^{2} satisfying f(x) = 0 in Z/p^{2}. Then 1 − μ(T_{f,M}) = ∏_{p≥M}(1 − c_{p}/p^{2}).

We are now ready to prove the analogue of Lemma 5.1 for squarefree values of multivariable polynomials over Z:

Lemma 6.2. Assume the abc conjecture. Suppose that f ∈ Z[x_{1}, …, x_{n}] is squarefree as a polynomial in Q[x_{1}, …, x_{n}], and suppose that x_{n} appears in each irreducible factor of f(x). Then lim_{M→∞} \overline{μ}_{n}(T_{f,M}) = 0.

Proof. Factors of f lying in Z are irrelevant as M → ∞, so we may assume that f is squarefree as a polynomial in Z[x_{1}, …, x_{n}]. If f factors as a product of two relatively prime polynomials g and h, then the result for f follows from the result for g and h together with Lemma 5.1 applied to g, h. Hence we may reduce to the case where f is irreducible in Z[x_{1}, …, x_{n}].

Let Δ ∈ Z[x_{1}, …, x_{n−1}] and δ ≥ 1 be the discriminant and degree, respectively, of f considered as a polynomial in x_{n}. Given B_{1}, …, B_{n}, let Q := T_{f,M} ∩ Box. We need to show that if the B_{i} are sufficiently large, and B_{n} is sufficiently large relative to the other B_{i}, then #Q/#Box is small.

The fraction of (a_{1}, …, a_{n−1}) in Box_{n−1} := Box(B_{1}, …, B_{n−1}) at which Δ vanishes is negligible, by Lemma 4.1. Since f is irreducible, when f is viewed as a polynomial in x_{n}, its coefficients (in Z[x_{1}, …, x_{n−1}]) are relatively prime (not necessarily pairwise). In particular, the common zero locus of these coefficients has codimension at least 2 in A_{Q}^{n−1}, hence is contained in the subvariety defined by f̃ = g̃ = 0 for two relatively prime elements f̃, g̃ ∈ Z[x_{1}, …, x_{n−1}]. Thus Lemma 5.1 implies that the fraction of (a_{1}, …, a_{n−1}) ∈ Box_{n−1} such that there exists a prime p ≥ M such that the image of f(a_{1}, …, a_{n−1}, x_{n}) in F_{p}[x_{n}] is zero is negligible, when M is large.

It remains to bound #(Q ∩ (Q′ × [1, B_{n}]))/#Box, where Q′ is the set of (a_{1}, …, a_{n−1}) ∈ Box_{n−1} such that

• Δ(a_{1}, …, a_{n−1}) ≠ 0, and

• there is no prime p ≥ M such that the image of f(a_{1}, …, a_{n−1}, x_{n}) in F_{p}[x_{n}] is zero.

By the first condition, Lemma 6.1 applies to f(a_{1}, …, a_{n−1}, x_{n}) ∈ Z[x_{n}] for each (a_{1}, …, a_{n−1}) ∈ Q′. Letting B_{n} tend to infinity while B_{1}, …, B_{n−1} are fixed, we find that it suffices to bound

(1)  \frac{1}{#Box} ∑_{(a_{1},...,a_{n−1})∈Q′} B_{n} (1 − ∏_{p≥M} (1 − \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}})) ,

where c_{p}(a_{1}, …, a_{n−1}) is the number of x_{n} ∈ Z/p^{2} such that f(a_{1}, …, a_{n−1}, x_{n}) = 0 in Z/p^{2}. The inequality 1 − αβ ≤ (1 − α) + (1 − β) holds for α, β ∈ [0, 1]; applying this with

α := ∏_{{p≥M \atop p∤Δ(a_{1},...,a_{n−1})}} (1 − \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}}) ,  β := ∏_{{p≥M \atop p|Δ(a_{1},...,a_{n−1})}} (1 − \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}})

and using

1 − α ≤ ∑_{{p≥M \atop p∤Δ(a_{1},...,a_{n−1})}} \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}}

bounds (1) by s_{1} + s_{2} where

s_{1} := \frac{1}{#Box} ∑_{(a_{1},...,a_{n−1})∈Q′} B_{n} ∑_{{p≥M \atop p∤Δ(a_{1},...,a_{n−1})}} \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}},

s_{2} := \frac{1}{#Box} ∑_{(a_{1},...,a_{n−1})∈Q′} B_{n} (1 − ∏_{{p≥M \atop p|Δ(a_{1},...,a_{n−1})}} (1 − \frac{c_{p}(a_{1}, …, a_{n−1})}{p^{2}})) .

When (a_{1}, …, a_{n−1}) ∈ Q′ and p ∤ Δ(a_{1}, …, a_{n−1}), Hensel’s Lemma implies c_{p}(a_{1}, …, a_{n−1}) ≤ δ, while B_{n}#Q′ ≤ #Box, so s_{1} ≤ ∑_{p≥M} δ/p^{2}, which is negligible as M → ∞. When (a_{1}, …, a_{n−1}) ∈ Q′ and p | Δ(a_{1}, …, a_{n−1}), the image of f(a_{1}, …, a_{n−1}, x_{n}) in F_{p}[x_{n}] has at most δ zeros in F_{p}; so c_{p}(a_{1}, …, a_{n−1}) ≤ δp, and

(2)  s_{2} ≤ \frac{1}{#Box} ∑_{(a_{1},...,a_{n−1})∈Q′} B_{n} (1 − ∏_{{p≥M \atop p|Δ(a_{1},...,a_{n−1})}} (1 − \frac{δ}{p})) .

Let Φ(x_{n}) = ∏_{j=1}^{δ}(x_{n} − j). We may assume M ≥ δ; then

1 − ∏_{{p≥M \atop p|Δ(a_{1},...,a_{n−1})}} (1 − \frac{δ}{p})

equals the density of

{ x_{n} ∈ Z : ∃p ≥ M such that p | Δ(a_{1}, …, a_{n−1}), Φ(x_{n}) }.

Hence, as B_{n} → ∞ for fixed M, B_{1}, …, B_{n−1}, the right hand side of (2) has the same limit as

\frac{#((Q′ × [1, B_{n}]) ∩ Q_{Δ(x_{1},...,x_{n−1}),Φ(x_{n}),M})}{#Box}.

The latter is negligible, by Lemma 5.1. □

Remark. It seems difficult to improve Lemma 6.2 to obtain a result for the more natural definition of density, \overline{μ} instead of \overline{μ}_{n}. This would require a version of Granville’s onevariable result that is uniform in the coefficients of the polynomial. Granville’s proof uses Belyi functions, however, whose degrees vary wildly with the coefficients.

Proof of Theorem 3.2. Approximate S_{f} by

S_{f,M} := { a ∈ Z^{n} : f(a) is not divisible by p^{2} for any prime p < M }.

Then S_{f,M} is a union of cosets of (IZ)^{n} where I = ∏_{p<M} p^{2}. The Chinese Remainder Theorem implies μ_{n}(S_{f,M}) = ∏_{𝔭∈P_{M}}(1 − c_{p}/p^{2n}). By Lemma 6.2,

μ_{n}(S_{f}) = lim_{M→∞} μ_{n}(S_{f,M}) = ∏_{p}(1 − c_{p}/p^{2n}).

Finally, we show that the product converges (instead of diverging to 0) by showing that c_{p} = O(p^{2n−2}). (The value of the product could still be zero if some factor were zero.) Let X be the subscheme of A_{Z}^{n} defined by f = 0. Since the field Q is perfect, the nonsmooth locus of X × Q → Spec Q has codimension at least 2 in A_{Q}^{n}. It follows that for sufficiently large p, the nonsmooth locus Y_{p} of X × F_{p} → Spec F_{p} has codimension at least 2 in A_{F_{p}}^{n}, so #Y_{p}(F_{p}) = O(p^{n−2}). Each point in Y_{p}(F_{p}) can be lifted to an n-tuple in (Z/p^{2})^{n} in p^{n} ways. On the other hand, #(X_{p} − Y_{p})(F_{p}) = O(p^{n−1}) but the nonvanishing of some derivative modulo p at a point in (X_{p} −Y_{p})(F_{p}) implies that such a point lifts to at most p^{n−1} solutions to f(x) = 0 in (Z/p^{2})^{n}. Thus c_{p} = p^{n}O(p^{n−2}) + p^{n−1}O(p^{n−1}) = O(p^{2n−2}). □

7. Squarefree values of polynomials over F_{q}[t]

Throughout this section, A = F_{q}[t] where q = p^{e}. Our goal is to prove Theorem 3.4. We begin by stating the analogue of Lemma 6.2.

Lemma 7.1. Suppose that f ∈ A[x_{1}, …, x_{n}] is squarefree as a polynomial in K[x_{1}, …, x_{n}]. Then lim_{M→∞} \overline{μ}(T_{f,M}) = 0.

Before beginning the proof of Lemma 7.1, we state and prove two results that will be needed in its proof.

Lemma 7.2. If f ∈ K[x_{1}, …, x_{n}] is squarefree, then

F := f (y_{0}^{p} + ty_{1}^{p} + ⋯ + t^{p−1}y_{p−1}^{p}, x_{2}, x_{3}, …, x_{n}) ∈ K[y_{0}, …, y_{p−1}, x_{2}, …, x_{n}]

is squarefree.

Proof. We work in B := K^{1/p}[y_{0}, …, y_{p−1}, x_{2}, …, x_{n}], where K^{1/p} = F_{q}(t^{1/p}). Define u := y_{0} + t^{1/p}y_{1} + ⋯ + t^{(p−1)/p}y_{p−1}.

We first show that K^{1/p}[u] ∩ K[y_{0}, …, y_{p−1}] = K[u^{p}]. Suppose g = ∑ α_{i}u^{i} ∈ K^{1/p}[u] ∩ K[y_{0}, …, y_{p−1}]. The coefficient of y_{0}^{i} in g is α_{i}, so α_{i} ∈ K for all i. The coefficient of y_{0}^{i−1}y_{1} in g is it^{1/p}α_{i}; this too is in K, so for all i, either p | i or α_{i} = 0. Thus g ∈ K[u^{p}], as desired.

It follows that

(3)  K^{1/p}[u, x_{2}, …, x_{n}] ∩ K[y_{0}, …, y_{p−1}, x_{2}, …, x_{n}] = K[u^{p}, x_{2}, …, x_{n}].

Suppose that G^{2} | F for some G ∈ K[y_{0}, …, y_{p−1}, x_{2}, …, x_{n}] − K. Then G^{2} | F in B. If we view B as a polynomial ring over K^{1/p} in algebraically independent indeterminates u, y_{1}, …, y_{p−1}, x_{2}, …, x_{n} (by eliminating y_{0}), then F = f(u^{p}, x_{2}, …, x_{n}) does not involve y_{1}, …, y_{p−1}, so G ∈ K^{1/p}[u, x_{2}, …, x_{n}] too. By (3), G ∈ K[u^{p}, x_{2}, …, x_{n}]. If we write G = g(u^{p}, x_{2}, …, x_{n}) where g ∈ K[x_{1}, …, x_{n}], then g^{2} | f and g ∉ K, contradicting the assumption that f is squarefree in K[x_{1}, …, x_{n}]. □

Lemma 7.3. Suppose f ∈ K[x_{1}^{p}, …, x_{n}^{p}] is squarefree as an element of K[x_{1}, …, x_{n}]. Then f and ∂f/∂t are relatively prime as elements of K[x_{1}, …, x_{n}].

Proof. The gcd g of f and ∂f/∂t in K[x_{1}^{p}, …, x_{n}^{p}] equals their gcd in K[x_{1}, …, x_{n}]. We may multiply g by an element of K^{∗} to assume that some coefficient of g equals 1. Write f = gh. Since f is squarefree, g and h are relatively prime in K[x_{1}, …, x_{n}]. But g divides \frac{∂f}{∂t} = g \frac{∂h}{∂t} + h\frac{∂g}{∂t}, so g divides ∂g/∂t. The total degree of ∂g/∂t is less than or equal to that of g, so ∂g/∂t = cg for some c ∈ K. Then each coefficient γ of g satisfies ∂γ/∂t = cγ. One of these coefficients is 1, so c = 0. Thus each coefficient γ is in K^{p}, so g = G^{p} for some G ∈ K[x_{1}, …, x_{n}]. But g | f, and f is squarefree, so g ∈ K. Hence f and ∂f/∂t are relatively prime. □

Proof of Lemma 7.1. We may assume that f is squarefree as an element of A[x_{1}, …, x_{n}]. Define

F := f (∑_{j=0}^{p−1} t^{j}y_{1j}^{p}, …, ∑_{j=0}^{p−1} t^{j}y_{nj}^{p}) ∈ K[…, y_{ij}, …]

By n applications of Lemma 7.2, F is squarefree. Let B_{ij} = (B_{i}/|t^{j}|)^{1/p}. As each y_{ij} ranges through elements of A satisfying |y_{ij}| ≤ B_{ij}, the n-tuples

(∑_{j=0}^{p−1} t^{j}y_{1j}^{p}, …, ∑_{j=0}^{p−1} t^{j}y_{nj}^{p})

exhaust the elements of Box, with each element appearing once. Hence Lemma 7.1 for f follows from Lemma 7.1 for F.

Evaluation of F at an element a ∈ A^{np} commutes with formal application of ∂/∂t, since F ∈ K[…, y_{ij}^{p}, …]. Thus, for a prime 𝔭 of A, we have 𝔭^{2} | F (a) if and only if 𝔭 divides F (a) and (∂F/∂t)(a). Hence Lemma 7.1 for F follows from Lemma 5.1 for F and ∂F/∂t, which are relatively prime by Lemma 7.3. □

Proof of Theorem 3.4. We mimic the proof of Theorem 3.2 at the end of Section 6, using Lemma 7.1 in place of Lemma 6.2. But the last paragraph of that proof, proving c_{p} = O(p^{2n−2}) to obtain convergence of the infinite product, does not carry over, since it uses the fact that Q is perfect to obtain generic smoothness, and F_{q}(t) is not perfect.

Therefore we prove c_{𝔭} = O(|𝔭|^{2n−2}) by a different method. The fraction c_{𝔭}/|𝔭|^{2n} is unchanged when we replace the c_{𝔭} and n for the original f by the corresponding values for the F defined in the beginning of the proof of Lemma 7.1. This fraction for F is bounded by the fraction of ā ∈ (A/𝔭)^{np} such that F and ∂F/∂t vanish mod 𝔭 at ā. By Lemma 7.3, F and ∂F/∂t are relatively prime in K[…, y_{ij}, …], so they define a subscheme of codimension at least 2 in A_{K}^{np}, and the desired bound follows, just as in the last two sentences of Section 5. □

8. Squarefree values of polynomials over other rings of functions

Let S be a finite nonempty set of closed points of a smooth, projective, geometrically integral curve X over F_{q}. Define the affine curve U := X \ S, and let A be the ring of regular functions on U. Thus A is the set of S-integers of the function field K of X. An element a ∈ A is called squarefree if the ideal (a) of A is a product of distinct primes of A. Let Div_{S} denote the set of effective divisors on X with support contained in S. If D ∈ Div_{S}, then the F_{q}-subspace L(D) := { f ∈ K^{∗} : (f) + D ≥ 0 } ∪ {0} of K is contained in A. Let ℓ(D) = dim_{F_{q}} L(D). Define the density μ(S) of a subset S ⊆ A^{n} as the limit of #(S ∩ Box)/#Box as Box runs through L(D_{1}) × ⋯ × L(D_{n}), where D_{i} ∈ Div_{S} and min_{i} deg D_{i} → ∞. The following theorem generalizes Theorem 1 in [Ram92]; we state it only for squarefree values, but as mentioned already in Section 3 the lemmas used in its proof will yield the analogous result for k^{th}-power-free values.

Theorem 8.1 (Squarefree values over rings of functions). With notation as above, let f ∈ A[x_{1}, …, x_{n}] be a polynomial that is squarefree as an element of K[x_{1}, …, x_{n}]. Let

S_{f} := { a ∈ A^{n} : f(a) is squarefree }.

For each nonzero prime 𝔭 ⊆ A, let c_{𝔭} be the number of x ∈ (A/𝔭^{2})^{n} satisfying f(x) = 0 in A/𝔭^{2}. Then μ(S_{f}) = ∏_{𝔭}(1 − c_{𝔭}/|𝔭|^{2n}), where |𝔭| := #(A/𝔭).

Sketch of proof of Theorem 8.1. Most of the proof follows the proofs of Theorems 3.1 and 3.4, so we will comment only on the differences.

First we prove the analogue of Lemma 5.1. Enlarging S only makes this analogue harder to prove, since there are more boxes to consider, while the primes 𝔭 involved are the same once M exceeds |𝔭| for every new 𝔭 thrown into S. Thus we may assume that A is a principal ideal domain. Let g be the genus of X. Given Box = L(D_{1}) × ⋯ × L(D_{n}), let B_{i} = deg D_{i} − (2g − 2). We may assume deg D_{1} ≤ ⋯ ≤ deg D_{n}. Replace the definition of Q_{s} in the proof of Lemma 5.1 by

Q_{s} := { a ∈ Box : ∃𝔭 such that B_{s} ≤ deg 𝔭 < B_{s+1} and 𝔭 | f(a), g(a) }.

Replace the argument “Since |𝔭| < B_{s+1}, the reductions modulo 𝔭 of the (a_{s+1}, …, a_{n}) ∈ Box(B_{s+1}, …, B_{n}) are almost uniformly distributed in (A/𝔭)^{n−s}” by “By Riemann-Roch, if D is a divisor with deg D −deg 𝔭 > 2g −2, then ℓ(D)−ℓ(D −𝔭) = deg 𝔭; hence the F_{q}-linear reduction map

L(D_{s+1}) × ⋯ × L(D_{n}) → (A/𝔭)^{n−s}

is surjective for deg 𝔭 < B_{s+1}.” Replace the estimate “|h(a_{1}, …, a_{s})| = O(B_{s}^{η})” by “deg h(a_{1}, …, a_{s}) = O(deg D_{s}) = O(B_{s}) where the implied constants depend on h and the genus g, but not on the a_{i} or D_{i}”

to bound the number of primes 𝔭 with deg 𝔭 ≥ B_{s} dividing h(a_{1}, …, a_{s}). The rest of the proof of Lemma 5.1 goes as before.

Next we prove the analogue of Lemma 7.1. Choose any t ∈ A−A^{p}, and note that Lemmas 7.2 and 7.3 go through without change; in fact, we can generalize Lemma 7.2 by replacing the form y_{0}^{p} + ty_{1}^{p} + ⋯ + t^{p−1}y_{p−1}^{p} by t_{1}y_{1}^{p} + t_{2}y_{2}^{p} + ⋯ + t_{m}y^{m} for any t_{1}, …, t_{m} ∈ A provided that t_{i}/t_{j} ∉ K^{p} for some i, j.

We claim that there exist t_{1}, …, t_{m} ∈ A such that for every D ∈ Div_{S} and every a ∈ L(D), it is possible to write a = t_{1}a_{1}^{p} + ⋯ + t_{m}a_{m}^{p} with a_{i} ∈ A such that t_{i}a_{i}^{p} ∈ L(D) for each i. More precisely, we claim that if E is the divisor (p + 1)(2g + 5) ∑_{𝔭∈S} 𝔭, then the choice {t_{1}, …, t_{m}} := L(E) works. To prove this, it suffices to show, given D ∈ Div_{S} and a ∈ L(D) − L(E), that one can adjust a by an element of the form t_{i}a_{i}^{p} in L(D) to obtain an element in L(D − 𝔭) for some 𝔭 in the support of D; then iterate. If a ∈ L(D) − L(E), then for some 𝔭 ∈ S, a has a pole of order greater than (p + 1)(2g + 5) at 𝔭. If n ≥ 2g + 4, Riemann-Roch shows that ℓ(n𝔭)−ℓ((n−1)𝔭) = deg 𝔭; in particular there exists a function in A having a pole of order n with prescribed leading coefficient at 𝔭, and no other poles. Since every integer greater than (p + 1)(2g + 5) is expressible as i + pj with 2g + 4 ≤ i < 2g + 4 + p and j ≥ 2g + 4, we can find functions t ∈ L(i𝔭) and α ∈ L(j𝔭) such that tα^{p} and a have the same order of pole at 𝔭, and the same leading coefficient. Then t ∈ L(E) and tα^{p} ∈ L(D), so this proves the claim.

By the previous paragraph, for each D ∈ Div_{S}, we have a surjection

L(D_{1}) × ⋯ × L(D_{r}) → L(D)
(a_{1}, …, a_{r}) ↦ t_{i_{1}}a_{1}^{p} + ⋯ + t_{i_{r}}a_{r}^{p}

for some subset {i_{1}, …, i_{r}} ⊆ {1, 2, …, m} and for some D_{i} ∈ Div_{S} depending on D. If deg D is sufficiently large, then L(D) contains nonzero functions with ratio outside K^{p}, so some t_{i_{α}}/t_{i_{β}} lies outside K^{p}. Thus in proving the analogue of Lemma 7.1, we may reduce the result for f to the result for each F in the finite set of polynomials of the form

F := f (∑_{j∈S_{1}} t_{j}y_{1j}^{p}, …, ∑_{j∈S_{n}} t_{j}y_{nj}^{p}) ∈ K[…, y_{ij}, …]

for all possible subsets S_{1}, …, S_{n} ⊆ {1, 2, …, m} such that each subset contains i, j with t_{i}/t_{j} ∉ K^{p}.

Choose b ∈ A such that b\frac{∂}{∂t} is a nonzero derivation A → A. Then the analogue of Lemma 7.1 for an F as above follows from the analogue of Lemma 5.1 applied to F and b∂F/∂t.

To complete the proof of Theorem 8.1, it remains to prove c_{𝔭} = O(|𝔭|^{2n−2}), to obtain convergence of the infinite product. This follows as in the proof of Theorem 3.4 at the end of Section 7, except that the fraction c_{𝔭}/|𝔭|^{2n} for f is now bounded by a sum of the analogous fractions for several different polynomials F. Each of the latter fractions is bounded by O(1/|𝔭|^{2}), and the number of polynomials F is O(1), independent of 𝔭, so c_{𝔭}/|𝔭|^{2n} = O(1/|𝔭|^{2}), as desired. □

9. The image of the values in Q^{∗}/Q^{∗2}

This section is devoted to the proof of Theorem 3.5.

Lemma 9.1. Suppose that f(x) ∈ Z[x] is squarefree as an element of Q[x], and that deg f ≥ 2. Fix q ∈ Q^{∗} such that q ≠ 1. Then the number of solutions (m, n) to f(m) = qf(n) satisfying 1 ≤ m, n ≤ B is o(B) as B → ∞.

Proof. Since f(m) has constant sign for large positive m, the result is trivial if q < 0. Therefore assume q > 0.

Theorem 2 in Chapter 13 of [Ser97] implies that the number of such solutions on each irreducible component of the curve f(m)−qf(n) = 0 in the (m, n)-plane over Q is O(B^{1/2} log B) unless some component is a line. If there is a line, it cannot be of the form n = α for any α ∈ Q, so it would have an equation m = αn + β for some α, β ∈ Q. Then f(αn + β) − qf(n) = 0 as polynomials in Q[n]. Equating leading coefficients shows that α ≠ 1. Choose γ ∈ Q so that αγ + β = γ. Then the polynomial F (x) := f(x + γ) satisfies F (αx) − qF (x) = 0. Since q > 0 and q ≠ 1, there is at most one integer d such that α^{d} = q. Thus F is a monomial of degree d, and f(n) = c(n − γ)^{d} for some c ∈ Q. Since deg f ≥ 2, this contradicts the assumption that f is squarefree. □

We now begin the proof of Theorem 3.5. We easily reduce to the case where f(x) = h(x), that is, where the coefficients of f have gcd 1, and f is squarefree in Z[x]. Also we may assume that the leading coefficient of f is positive. The deg f = 0 case is trivial.

Proof of Theorem 3.5 in the case deg f = 1. Write f(x) = ax + b. Changing b by a multiple of a changes the sequence of values only in finitely many terms, so we may assume 0 < b ≤ a. Given r, N ∈ Z_{≥0}, let S(r mod a, N) denote the set of positive squarefree integers ≤ N that are congruent to r modulo a. Identify each element of Q^{∗}/Q^{∗2} with a squarefree integer representative.

We claim that the image of {f(1), f(2), …, f(B)} in Q^{∗}/Q^{∗2} is the disjoint union of S(r mod a, δ_{r}(aB + b)) as r ranges from 0 to a − 1. Let m be as in the definition of δ_{r}, when it exists. If s ∈ S(r mod a, δ_{r}(aB + b)), then δ_{r} > 0, so the integer m in the definition of δ_{r} exists; then m^{2}s ≡ m^{2}r ≡ b (mod a) and m^{2}s ≤ m^{2}δ_{r}(aB + b) = aB + b, so m^{2}s ∈ {f(1), f(2), …, f(B)}, and hence s is in the image. Conversely, suppose that the squarefree integer s represents the image of f(n) in Q^{∗}/Q^{∗2} for some n ∈ {1, 2, …, B}. Thus f(n) = m̄^{2}s for some m̄. Let r ∈ [0, a) be such that r ≡ s (mod a). Then m̄^{2}r ≡ m̄^{2}s = an + b ≡ b (mod a), so the m in the definition of δ_{r} exists, and m ≤ m̄. Now s = f(n)/m̄^{2} ≤ (aB + b)/m^{2} = δ_{r}(aB + b), so s ∈ S(r mod a, δ_{r}(aB + b)). This proves the claim.

When gcd(a, r) = 1, the density of squarefree values of ax + r equals

∏_{p∤a}(1 − p^{−2}) = \frac{6}{π^{2}} ∏_{p|a} (1 − p^{−2})^{−1} ,

so

#S(r mod a, N) = (\frac{6}{π^{2}} ∏_{p|a} (1 − p^{−2})^{−1} + o(1)) (N/a)

as N → ∞. The result follows upon setting N = δ_{r}(aB + b) ∼ δ_{r}aB for each r ∈ [0, a) for which δ_{r} ≠ 0 (such r are necessarily prime to a), and summing over r. □

Proof of Theorem 3.5 in the case deg f ≥ 2. Replacing f(x) by f(x+n) for some n, we may assume that 0 < f(1) < f(2) < · · ·. It suffices to show, given ϵ > 0, that for sufficiently large B, the image of {f(1), …, f(B)} in Q^{∗}/Q^{∗2} has size at least (1 − ϵ)B. For a positive integer n, let 𝔰(n) denote the largest positive integer m such that m^{2} | n. Define

S_{m} = { n ∈ {1, 2, …, B} : 𝔰(f(n)) = m }.

By Lemma 6.1, the density of the set of integers n such that 𝔰(f(n)) is divisible by a prime p > M tends to zero as M → ∞. Also, for each fixed prime ℓ, the set of integers n such that ℓ^{m} | 𝔰(f(n)) tends to zero as m → ∞, because the number of n ∈ Z/ℓ^{2m} such that f(n) = 0 in Z/ℓ^{2m} is O(1) as m → ∞, since f is squarefree. Hence μ ({ n ∈ Z : 𝔰(f(n)) ≥ M }) → 0 as M → ∞. In particular, if M is sufficiently large, then #(S_{1} ∪ ⋯ ∪ S_{M−1}) > (1 − ϵ/2)B for large B.

Now f maps each S_{i} injectively into Q^{∗}/Q^{∗2}. For 1 ≤ i < j < M, the intersection of the images of f(S_{i}) and f(S_{j}) in Q^{∗}/Q^{∗2} has size o(B) as B → ∞ by Lemma 9.1. Thus the image of \{f(1), f(2), …, f(B)\} in Q^{∗}/Q^{∗2} has size at least (1 − ϵ/2)B − \binom{M−1}{2}o(B) > (1 − ϵ)B, if B is sufficiently large relative to M. □

Acknowledgements

I thank Johan de Jong and Ofer Gabber for independently suggesting that I try to use ideas from [Poo02] to compute the density of squarefree values in the F_{q}[t] case. I thank Brian Conrad for pointing out some errors in an earlier draft, and the referee for helpful suggestions. Finally, I thank Pamela Cutter, Andrew Granville, and Tom Tucker for sharing their preprint with me.

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: poonen@math.berkeley.edu