POLYNOMIAL PARAMETRIZATION OF THE SOLUTIONS OF DIOPHANTINE EQUATIONS OF GENUS 0

SOPHIE FRISCH AND GÜNTER LETTL

Dedicated to Prof. Władysław Narkiewicz on the occasion of his 70^{th} birthday.

Abstract. Let f ∈ ℤ[X, Y, Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f = 0 has a function field isomorphic to the rational function field ℚ(T). We show that all integral solutions of the Diophantine equation f = 0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients.

Recently, the first author and L. Vaserstein proved that the set of all Pythagorean triples can be parametrized by a single triple of integer-valued polynomials, but not by a single triple of polynomials with integer coefficients (in any number of variables) [2]. We denote by Int (ℤ^{m}) the ring of integer-valued polynomials in m variables,

Int (ℤ^{m}) = {φ ∈ ℚ[X_{1}, …, X_{m}] | φ(ℤ^{m}) ⊂ ℤ} .

In this paper we will generalize the affirmative part of [2] to such homogeneous equations as define a (plane) projective curve with a rational function field.

Throughout this paper, f ∈ ℤ[X, Y, Z] \ {0} denotes an irreducible polynomial with integer coefficients, which is homogeneous of degree n ≥ 1. Let \overline{ℚ} be an algebraic closure of ℚ and C_{f} ⊂ ℙ^{2}(\overline{ℚ}) the plane projective curve defined by f = 0,

C_{f} = \{(x : y : z) ∈ ℙ^{2}(\overline{ℚ}) | f(x, y, z) = 0\} .

We will further suppose that the function field K = ℚ(C_{f}) of C_{f} over ℚ is isomorphic to the rational function field ℚ(T). This implies that f is absolutely irreducible (i.e., irreducible in \overline{ℚ}[X, Y, Z]). Our assumption is satisfied, for instance, if C_{f} has genus 0 and possesses a regular point defined over ℚ.

Recall that a point (x : y : z) ∈ C_{f} is singular if and only if the local ring R_{(x:y:z)} ⊂ K of all rational functions of C_{f} that are defined at (x : y : z) is not a discrete valuation ring (cf. [3, pp. 56-57]). In this case, there are finitely many discrete valuation rings O_{P_{i}} ⊂ K above R_{(x:y:z)} (meaning R_{(x:y:z)} ⊂ O_{P_{i}} and 𝔪_{(x:y:z)} ⊂ P_{i}, where 𝔪_{(x:y:z)} and P_{i} denote the corresponding maximal ideals). Let C_{f}^{bad} denote the set of those singular points (x : y : z) ∈ C_{f} for which there exists no discrete valuation ring O_{P} above R_{(x:y:z)} with O_{P} /P ≃ ℚ. These points will be “bad” for our main theorem.

We investigate the set of integer solutions of the Diophantine equation f(X, Y, Z) = 0,

L_{f} := {(x, y, z) ∈ ℤ^{3} | f(x, y, z) = 0} ,

2000 Mathematics Subject Classification. Primary 11D85, secondary 13F20, 11D41, 14H05.

up to those solutions which correspond to the “bad” points of the curve. We set

L_{f}^{bad} = {(x, y, z) ∈ L_{f} | (x : y : z) ∈ C_{f}^{bad}} .

Theorem 1. Let f ∈ ℤ[X, Y, Z]\{0} be an irreducible, homogeneous polynomial of degree n ≥ 1 such that the function field K = ℚ(C_{f}) is isomorphic to ℚ(T). Then there exist polynomials g_{1}, g_{2}, g_{3} ∈ Int (ℤ^{m}) for some m ∈ ℕ such that

L_{f} \ L_{f}^{bad} = \{(g_{1}(\underline{x}), g_{2}(\underline{x}), g_{3}(\underline{x})) | \underline{x} ∈ ℤ^{m}\};

in other words, up to the “bad” solutions, all solutions of the Diophantine equation

(1)  f(X, Y, Z) = 0

can be parametrized by one triple of integer-valued polynomials.

The suppositions of Theorem 1 imply that for n ≤ 2 the curve C_{f} has no singular point. For n = 1, C_{f} is just a line and the result of Theorem 1 is obvious (even with g_{i} ∈ ℤ[U, V]). For n = 2, we immediately obtain

Corollary 2. Let f ∈ ℤ[X, Y, Z] be an absolutely irreducible quadratic form. Then there exist polynomials g_{1}, g_{2}, g_{3} ∈ Int (ℤ^{m}) for some m ∈ ℕ such that

L_{f} = \{(g_{1}(\underline{x}), g_{2}(\underline{x}), g_{3}(\underline{x})) | \underline{x} ∈ ℤ^{m}\} .

For the proof of Theorem 1 we will use the resultant of polynomials and therefore recall some well-known results on it (cf. [5, Chap. I, §9-10]). Given polynomials g, h ∈ ℤ[U, V] in the variables U, V, let Res_{V} (g, h) ∈ ℤ[U] denote the resultant of g, h when considered as polynomials in the variable V over the ring ℤ[U], and, vice versa, Res_{U} (g, h) ∈ ℤ[V] the resultant of g, h as polynomials in U.

Lemma 3. Let g, h ∈ ℤ[U, V] be relatively prime polynomials.

a) Then Res_{U} (g, h) ≠ 0 and Res_{V} (g, h) ≠ 0, and there exist polynomials r, s, r′, s′ ∈ ℤ[U, V] with

gr + hs = Res_{U} (g, h) and gr′ + hs′ = Res_{V} (g, h) .

b) If g and h are homogeneous of degree d_{1} and d_{2}, resp., then Res_{U} (g, h) and Res_{V} (g, h) are each homogeneous of degree d_{1}d_{2}, and consequently

Res_{U} (g, h) = a V^{d_{1}d_{2}} and Res_{V} (g, h) = b U^{d_{1}d_{2}} with a, b ∈ ℤ \ {0} .

We will also use the implication (D)⇒(B) of the main theorem of [1], which for the sake of completeness we state in the following

Proposition 4. Let k ∈ ℕ and suppose that S ⊂ ℤ^{k} is the set of integer k-tuples in the range of a k-tuple of polynomials with rational coefficients, as the variables range through the integers, i.e., there exist h_{1}, …, h_{k} ∈ ℚ[X_{1}, …, X_{r}] for some r ∈ ℕ such that

S = \{(h_{1}(\underline{x}), …, h_{k}(\underline{x})) | \underline{x} ∈ ℤ^{r}\} ∩ ℤ^{k} .

Then S is parametrizable by a k-tuple of integer-valued polynomials, i.e., there exist g_{1}, …, g_{k} ∈ Int (ℤ^{m}) for some m ∈ ℕ such that

S = \{(g_{1}(\underline{x}), …, g_{k}(\underline{x})) | \underline{x} ∈ ℤ^{m}\} .

POLYNOMIAL PARAMETRIZATION OF THE SOLUTIONS OF DIOPHANTINE EQUATIONS 3

Proof of Theorem 1. Let f be as in the statement of the theorem. Then there exist homogeneous polynomials h_{1}, h_{2}, h_{3} ∈ ℚ[U, V] such that

(X, Y, Z) = (h_{1}(U, V), h_{2}(U, V), h_{3}(U, V))

defines a birational (projective) isomorphism between C_{f} and the projective line. We may assume h_{1}, h_{2}, h_{3} ∈ ℤ[U, V] and gcd(h_{1}, h_{2}, h_{3}) = 1 (see, for instance, [4, Sect. 2]).

For every ℚ-rational point (u : v) ∈ ℙ^{1}(ℚ), (h_{1}(u, v) : h_{2}(u, v) : h_{3}(u, v)) is the evaluation of the birational isomorphism at this point. This means that (h_{1}(u, v) : h_{2}(u, v) : h_{3}(u, v)) is a ℚ-rational point of C_{f} and its local ring is contained in some discrete valuation ring of K of degree 1. Therefore

L_{ℚ} := {(w h_{1}(u, v), w h_{2}(u, v), w h_{3}(u, v)) | u, v, w ∈ ℚ} =

{(w h_{1}(u, v), w h_{2}(u, v), w h_{3}(u, v)) | w ∈ ℚ, u, v ∈ ℤ with gcd(u, v) = 1}

is exactly the set of all rational solutions of (1) except for those corresponding to points of C_{f}^{bad}, and L_{f} \ L_{f}^{bad} = L_{ℚ} ∩ ℤ^{3} is just the set of all integral triples of L_{ℚ}.

We claim that there exists some d ∈ ℕ such that for all u, v ∈ ℤ with gcd(u, v) = 1 it follows that

gcd(h_{1}(u, v), h_{2}(u, v), h_{3}(u, v)) | d .

Let gcd(h_{1}, h_{2}) = t ∈ ℤ[U, V] and put h_{i} = t h′_{i} with h′_{i} ∈ ℤ[U, V], i = 1, 2. Since h′_{1}, h′_{2} are relatively prime, we obtain that Res_{V} (h′_{1}, h′_{2}) = a U^{δ} with some 0 ≠ a ∈ ℤ and δ ≥ 0, and polynomials ρ_{1}, ρ_{2} ∈ ℤ[U, V] with ρ_{1}h_{1} + ρ_{2}h_{2} = atU^{δ}. Since h_{1}, h_{2}, h_{3} were assumed to be relatively prime, gcd(atU^{δ}, h_{3}) = cU^{α} with c ∈ ℤ and 0 ≤ α ≤ δ. Dividing both atU^{δ} and h_{3} by cU^{α} and applying the same reasoning as above we finally obtain that there are 0 ≠ a_{1} ∈ ℤ, δ_{1} ≥ 0 and polynomials φ_{1}, φ_{2}, φ_{3} ∈ ℤ[U, V] with

(2)  φ_{1}h_{1} + φ_{2}h_{2} + φ_{3}h_{3} = a_{1}U^{δ_{1}} .

Using Res_{U} in the same way, we obtain polynomials ψ_{1}, ψ_{2}, ψ_{3} ∈ ℤ[U, V], 0 ≠ a_{2} ∈ ℤ and δ_{2} ≥ 0 such that

(3)  ψ_{1}h_{1} + ψ_{2}h_{2} + ψ_{3}h_{3} = a_{2}V^{δ_{2}} .

For any u, v ∈ ℤ with gcd(u, v) = 1, (2) and (3) imply that gcd(h_{1}(u, v), h_{2}(u, v), h_{3}(u, v)) divides both a_{1}u^{δ_{1}} and a_{2}v^{δ_{2}}. It follows that

gcd(h_{1}(u, v), h_{2}(u, v), h_{3}(u, v)) | lcm(a_{1}, a_{2}) := d .

So we obtain polynomials k_{i} = \frac{1}{d}h_{i} ∈ ℚ[U, V] with rational coefficients such that

L_{f} \ L_{f}^{bad} = {(w k_{1}(u, v), w k_{2}(u, v), w k_{3}(u, v)) | u, v, w ∈ ℤ} ∩ ℤ^{3} .

Now we apply Proposition 4, which yields the assertion of Theorem 1. □

Remarks. If the integers a_{1}, a_{2} appearing in (2) and (3) in the proof of Theorem 1 are both equal to 1, then k_{i} = h_{i} ∈ ℤ[U, V] and L_{f} \ L_{f}^{bad} can actually be parametrized by a triple of polynomials with integral coefficients (compare Example 2 below). When applying Proposition 4, we have no information about the number m of variables of the integer-valued polynomials g_{i} appearing in Theorem 1.

Example 1. This example shows that for n ≥ 3 “bad” singular points may appear. Consider

f = X^{3} + Y^{3} + X^{2}Z − 2Y^{2}Z ∈ ℤ[X, Y, Z] .

Then (0 : 0 : 1) ∈ C_{f} is a singular point. Only one discrete valuation ring lies over the local ring R_{(0:0:1)}, and this valuation ring has residue class field isomorphic to ℚ(\sqrt{2}). A birational (projective) isomorphism between C_{f} and the projective line is given by

(X : Y : Z) = ((V (2U^{2} − V^{2})) : (U(2U^{2} − V^{2})) : (V^{3} + U^{3})) ,

but there is no ℚ-rational point (u : v) ∈ ℙ^{1}(ℚ) corrsponding to the singular point (0 : 0 : 1). Indeed, the corresponding point (u : v) = (1 : \sqrt{2}) is only defined over ℚ(\sqrt{2}).

Example 2. In contrast to the Pythagorean triples (corresponding to the unit circle, see [2]), we know that for the equilateral hyperbola the set L_{f} can be parametrized by a single triple of polynomials with integer coefficients. Let

f = XY − Z^{2} ∈ ℤ[X, Y, Z] .

All ℚ-rational points of C_{f} are given by (u^{2} : v^{2} : uv) with (u : v) ∈ ℙ^{1}(ℚ). If u, v ∈ ℤ with gcd(u, v) = 1 then also gcd(u^{2}, v^{2}, uv) = 1. So the set of all integral solutions of XY − Z^{2} = 0 is given by

{(u^{2}w, v^{2}w, uvw) | u, v, w ∈ ℤ} .

References

[1] S. Frisch, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra (to appear).

[2] S. Frisch and L. Vaserstein, Parametrization of Pythagorean triples by a single triple of polynomials, J. Pure Appl. Algebra 212 (2008), 271-274.

[3] E. Kunz, Introduction to Plane Algebraic Curves, Birkhäuser, 2005.

[4] D. Poulakis and E. Voskos, Solving genus zero Diophantine equations with at most two infinite valuations, J. Symbolic Computation 33 (2002), 479-491.

[5] R.J. Walker, Algebraic Curves, Springer, 1978.

Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, AUSTRIA

  E-mail address: frisch@blah.math.tu-graz.ac.at

Institut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universität, Heinrichstraße 36, A-8010 Graz, AUSTRIA

  E-mail address: guenter.lettl@uni-graz.at