Polynomial Parametrization for the Solutions of Diophantine Equations and Arithmetic Groups

ANNALS OF MATHEMATICS

Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups

By Leonid Vaserstein

SECOND SERIES, VOL. 171, NO. 2 March, 2010

anmaah Annals of Mathematics, 171 (2010), 979–1009

Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups

By LEONID VASERSTEIN

Abstract

A polynomial parametrization for the group of integer two-by-two matrices with determinant one is given, solving an old open problem of Skolem and Beurk- ers. It follows that, for many Diophantine equations, the integer solutions and the primitive solutions admit polynomial parametrizations.

Introduction

This paper was motivated by an open problem from[8, p. 390]: CNTA 5.15 (Frits Beukers). Prove or disprove the following statement: There exist four polynomials A; B; C; D with integer coefficients (in any number of variables) such that AD BC 1 and all integer solutions D of ad bc 1 can be obtained from A; B; C; D by specialization of the D variables to integer values. Actually, the problem goes back to Skolem[14, p. 23]. Zannier[22] showed that three variables are not sufficient to parametrize the group SL2 ޚ, the set of all integer solutions to the equation x1x2 x3x4 1. D Apparently Beukers posed the question because SL2 ޚ (more precisely, a congruence subgroup of SL2 ޚ) is related with the solution set X of the equation x2 x2 x2 3, and he (like Skolem) expected the negative answer to CNTA 1 C 2 D 3 C 5.15, as indicated by his remark[8, p. 389] on the set X: I have begun to believe that it is not possible to cover all solutions by a finite number of polynomials simply because I have never

The paper was conceived in July of 2004 while the author enjoyed the hospitality of Tata Institute for Fundamental Research, India.

979 980 LEONID VASERSTEIN

seen a polynomial parametrisation of all two-by-two determinant one matrices with integer entries. In this paper (Theorem 1 below) we obtain the afﬁrmative answer to CNTA 5.15. As a consequence we prove, for many polynomial equations, that either the set X of integer solutions is a polynomial family or (more generally) X is a ﬁnite union of polynomial families. It is also possible to cover all solutions of x2 x2 x2 3 by two polynomial triples; see Example 15 below. 1 C 2 D 3 C Skolem[14, Bemerkung 1, p. 23] conjectured that SLn ޚ does not admit a polynomial parametrization for any n. However the main result of Carter and Keller[4] refutes this for n 3, and our Theorem 1 refutes this for n 2 and also D implies a similar result for n 3; see Corollary 17(a). A few words about our terminology. Let

.P1.y1; : : : ; yN /; : : : ; Pk.y1; : : : ; yN // be a k-tuple of polynomials in N variables with integer coefficients. Plugging in all N -tuples of integers, we obtain a family X of integer k-tuples, which we call a polynomial family (defined over the integers ޚ ) with N parameters. We also say that the set X admits a polynomial parametrization with N parameters. In other words, a polynomial family X is the image (range) P.ޚN / of a polynomial map P ޚN ޚk. We call this map P a polynomial parametrization of X. W ! Given a Diophantine equation or a system of Diophantine equations, we can ask whether the solution set (over ޚ) is a polynomial family. In other words, we can search for a general solution (i.e., a polynomial parametrization for the set). In the case of a polynomial equation, the polynomials in any polynomial parametrization form a polynomial solution. If no polynomial parametrization is known or exists, we can ask whether the solution set is a finite union of polynomial families. Loosely speaking, are the solutions covered by a finite number of polynomials? Also we can ask about polynomial parametrization of all primitive solutions. Recall that a k-tuple of integers is called primitive (or unimodular) if its GCD is 1. For any homogeneous equation, a polynomial parametrization of all primitive solutions leads to a polynomial parametrization of all solutions with one additional parameter. The open problem CNTA 5.15 quoted above is the question whether the group SL2 ޚ is a polynomial family, i.e., admits a polynomial parametrization. Our answer is “yes”:

THEOREM 1.SL 2 ޚ is a polynomial family with 46 parameters. We will prove this theorem in Section 1. The proof reﬁnes computations in [10],[2],[13],[18] and[4], especially, the last two papers. POLYNOMIAL PARAMETRIZATION 981

Now we consider some applications of the theorem and some examples. First we deal with an arbitrary system of linear equations. Then we consider quadratic equations. Finally, we consider Diophantine equations of higher degree. On the way, we make a few general remarks on the polynomial families. It is easy to see that the solution set for any system of linear equations (with integer coefﬁcients) either is empty or admits a polynomial parametrization of degree 1 with the number of parameters N less than or equal to the number of Ä variables k. In Section 2, using our Theorem 1, we obtain this: COROLLARY 2. The set of all primitive solutions for any linear system of equations with integer coefﬁcients either consists of 2 solutions or is a polyno- Ä mial family.

For example, the set Umn ޚ of all primitive (unimodular) n-tuples of integers turns out to be a polynomial family provided that n 2. The case n 3 is much easier, and this result can be easily extended to more general rings; see Section 2 below. When n 1, the set Um1 ޚ 1 consists of two elements. This set is D D f˙ g not a polynomial family but can be covered by two (constant) polynomials. In general, a finite set with cardinality 1 is not a polynomial family but can ¤ be covered by a finite number of (constant) polynomials (the number is zero in the case of an empty set). The set Umn ޚ is a projection of the set X of all integer solutions to the quadratic equation x1x2 x2n 1x2n 1. So if X is a polynomial family, C C D then obviously Umn ޚ is a polynomial family. Using Theorem 1, we will show X is a polynomial family if n 2. (When n 1 the solution set Um1 ޚ GL1 ޚ 1 D D D f˙ g to the equation x1x2 1 is not a polynomial family.) D COROLLARY 3. When n 2, the set of all integer solutions of x1x2 x2n 1x2n 1 C C D is a polynomial family. In fact, Theorem 1 implies that for many other quadratic equations, the set of all integer or all primitive solutions is a polynomial family or a finite union of polynomial families. A useful concept here is that of a Q-unimodular vector x, where Q.x/ is a quadratic form, i.e., a homogeneous, degree two polynomial with integer coefficient. An integer vector x is called Q-unimodular if there is a vector x such that Q.x x / Q.x/ Q.x / 1. Our Corollary 3 is a particular case 0 C 0 0 D of the following result, which we will prove in Section 3: COROLLARY 4. Consider the set X of all Q-unimodular solutions to Q.x/ D Q0, where Q.x/ is a quadratic form in k variables and Q0 is a given number. Assume that k 4 and that Q.x/ x1x2 x3x4 Q .x5; : : : ; x /. Then X is a D C C 0 k polynomial family with 3k 80 parameters. C 982 LEONID VASERSTEIN

Under the additional condition that k 6 and Q .x5; : : : ; x / x5x6 0 k D C Q .x7; : : : ; x /, it is easy to get a better bound (with 3k 6 instead of 3k 80) 00 k C without using Theorem 1 (see Proposition 3.4 below). Note that for nonsingular quadratic forms Q (when the corresponding symmetric bilinear form .x; x /Q Q.x x / Q.x/ Q.x / has an invertible matrix), 0 D C 0 0 “Q-unimodular” means “primitive.” In general, the orthogonal group acts on the integer solutions Z and on the Q-unimodular solutions X, a fact we use to prove Corollary 4. Now we make several remarks about the integer solutions x to Q.x/ Q0 that D are not Q-unimodular. We observe that both GCD.x/ and the GCD of all .x; x0/Q are invariants under the action. In the case when Q is nonsingular, Corollary 4 describes the set Z of all integer solutions z as follows: if Q0 0 (homogeneous D case), then z xy0 with primitive x, so Z is a polynomial family with an additional D parameter; if Q0 0, then Z is a ﬁnite union of polynomial families indexed ¤ by the square divisors of Q0. We will show elsewhere that the set Z for Q in Corollary 4 or, more generally, for any isotropic quadratic form Q, is a ﬁnite union of polynomial families. 2 Example 5. The solution set for the Diophantine equation x1x2 x admits D 3 a polynomial parametrization with three parameters: 2 2 .x1; x2; x3/ y1.y ; y ; y2y3/: D 2 3 Among these solutions, the primitive solutions are those with y1 1 and .y2; y3/ D ˙ Um2 ޚ. So by Theorem 1 (or Corollary 3 with n 2/, the set of all primitive 2 D solutions is the union of two polynomial families. The set of primitive solutions is not a polynomial family. This follows easily from the fact that the polynomial ring ޚŒy1; : : : ; yN is a unique factorization domain from any N , so within any polynomial family either all x1 0 or all x1 0. Ä The number 2 here is related to the fact that the group SL2 ޚ acts on the two- by-two symmetric matrices with two orbits on the determinant 0 primitive matrices. The action is Â Ã Â Ã x1 x3 T x1 x3 ˛ ˛ for ˛ SL2 ޚ, x3 x4 7! x3 x4 2 where ˛T is the transpose of ˛. An alternative description of the action of SL2 ޚ is Â Ã Â Ã x3 x1 1 x3 x1 ˛ ˛ for ˛ SL2 ޚ. x2 x3 7! x2 x3 2 2 The trace 0 and the determinant x1x2 x are preserved under this action. 3 POLYNOMIAL PARAMETRIZATION 983

Example 6. The solution set for x1x2 x3x4 0 admits the following poly- C D nomial parametrization with ﬁve parameters:

.x1; x2; x3; x4/ y5.y1y2; y3y4; y1y3; y2y4/: D Such a solution is primitive if and only if y5 1 and .y1; y4/; .y2; y3/ Um2 ޚ. D ˙ 2 So by Theorem 1, the set of all primitive solutions is a polynomial family with 92 parameters. By Theorem 2.2 below, the number of parameters can be reduced to 90. 2 Example 7. Consider the equation x1x2 x D with a given D ޚ. The D 3 C 2 case D 0 was considered in Example 5, so assume now that D 0. We can D ¤ identify solutions with integer symmetric two-by-two matrices of determinant D. The group SL2 ޚ acts on the set X of all solutions as in Example 5. It is easy to see and well known that every orbit contains either a matrix a b with a 0 and b d ¤ .1 a /=2 b a =2 d =2 or a matrix 0 b with b2 D. In the first case, j j Ä Ä j j Ä j j b 0 D D ad b2 a2 a2=4 3a2=4 3b2=16, and d is determined by a and b; j j D j j D hence the number of orbits is at most 8 D =3. Therefore the total number of orbit j j is bounded by 8 D =3 2. (Better bounds and connections with the class number j j C of the field ޑŒpD are known.) By Theorem 1, every orbit is a polynomial family with 46 parameters, so the set X can be covered by a finite set of polynomials, and the subset of primitive solutions can be covered by a finite set of polynomials with 46 parameters each. When D 1 or 2, the number of orbits and hence the number of polynomial D ˙ ˙ families is two. When D 3, the number of orbits is four. D ˙ When D is square-free, every integer solution is primitive.

Example 8. Consider the equation x1x2 x3x4 D with a given integer D C D (i.e., the equation in Corollary 4 with k 4 and Q0 D/. The case D 0 was D D D considered in Example 6, so assume now that D 0. The group SL2 ޚ SL2 ޚ ¤ acts on the solutions x1 x3 by x4 x2 Â Ã Â Ã x1 x3 1 x1 x3 ˛ ˇ; where ˛; ˇ SL2 ޚ. x4 x2 7! x4 x2 2 It is well known that every orbit contains the matrix diag.d; D=d/, where 2 d GCD.x1; x2; x3; x4/. So the number of orbits is the number of squares d D dividing D. By Theorem 1, the set X of integers solutions is a ﬁnite union of polynomial families, and the subset X 0 of primitive solutions is a polynomial family with 92 parameters. When D is square-free, X X. When D 1, Theorem 1 0 D D ˙ gives a better number of parameters, namely 46 instead of 92. Example 9. Let D 2 be a square-free integer. It is convenient to write 2 2 solutions .x1; x2/ .a; b/ of Pell’s equation x Dx 1 as a bpD ޚŒpD. D 1 2 D C 2 984 LEONID VASERSTEIN

Then they form a group under multiplication. All solutions are primitive, and they are parametrized by two integers, m and n, as m n a bpD .a0 b0pD/ .a1 b1pD/ ; C D C C where a0 b0pD is a solution of infinite order and a1 b1pD is a solution of C C finite order (this is not a polynomial parametrization!). We claim that every polynomial solution to the equation is constant. Since D is not a square, this is obvious. Here is a more sophisticated argument which works for many “sparse” sequences: It is clear that P a " < for any " > 0, where the sum is taken over all j j 1 solutions a bpD. On the other hand, if we have a nonconstant polynomial C solution, we have a nonconstant univariate solution f .y/ g.y/pD. If g.x/ is C not constant, then f .y/ is not constant. Let d 1 be the degree of f .x/. Then P f .z/ " provided that 0 < " 1=d. Since f .z/ takes every value at z ޚj j D 1 Ä most2 d times, we obtain a contradiction that proves d 0. D Since the set X of all integer solutions is infinite, it cannot be covered by a finite number of polynomials.

Remarks. (1) Let a1; a2;::: be a sequence of integers satisfying a linear recurrence equation an c1an 1 ckan k with some k 1 and ci ޚ for all D C C 2 n k 1. Then the argument in Example 9 shows that the set X of all integers ai C either is finite or is not a finite union of polynomial families. Note that X is finite if and only if any of the following conditions holds: the sequence is bounded, the sequence is periodic, the sequence satisfies a linear recurrence equation with all zeros of the char- acteristic polynomial being roots of 1, the sequence satisfies a linear recurrence equation with all zeros of the char- acteristic polynomial on the unit circle. (2) The partition function p.n/ provides another set of integers that is not a finite union of polynomial families (use the well-known asymptotic for p.n/ and the argument in Example 9). (3) By author’s request, S. Frisch[6] proved that every subset of ޚk with a finite complement is a polynomial family. (4) The set of all positive composite numbers is parametrized by the polynomial .y2 y2 y2 y2 2/.y2 y2 y2 y2 2/: 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C (5) It is known[9] that the union of the set of (positive) primes and a set of negative integers is a polynomial family. In the terms of[9], the set of primes is POLYNOMIAL PARAMETRIZATION 985 both Diophantine and listable. On the other hand, it is an easy fact that the set of primes is not a polynomial family; see Corollary 5.15 below and the remark after its proof. Corollaries 2–4 and Examples 5–9 above are about quadratic equations. The next three examples are about higher degree polynomial equations. Example 10. The Fermat equation yn yn yn with any given n 3 has 1 C 2 D 3 three “trivial” polynomial families of solutions with one parameter each when n is odd, and it has four polynomial families of solutions when n is even. Fermat’s Last Theorem says that these polynomial families cover all integer solutions. Example 11. It is unknown whether the solution set of x3 x3 x3 x3 0 1 C 2 C 3 C 4 D can be covered by a finite set of polynomials. A negative answer was conjectured in[7]. Example 12. It is unknown whether the solution set of x3 x3 x3 1 1 C 2 C 3 D can be covered by a finite number of polynomials. It is known (see[11, Th. 2]) that the set cannot by covered by a finite number of univariant polynomials. To deal with equations x2 x2 x2 and x2 x2 x2 3 (which are equiva- 1 C 2 D 3 1 C 2 D 3 C lent over the rational numbers ޑ to the equations in Examples 5 and 7 with D 3, D respectively) we need a polynomial parametrization of a congruence subgroup of SL2 ޚ. Recall that for any nonzero integer q, the principal congruence subgroup 1 0 SL2.qޚ/ consists of ˛ SL2 ޚ such that ˛ 12 modulo q.A congruence 2 Á D 0 1 subgroup of SL2 ޚ is a subgroup containing a principal congruence subgroup.

THEOREM 13. Every principal congruence subgroup of SL2 ޚ admits a polynomial parametrization with 94 parameters. We will prove this theorem in Section 5 below. Theorem 13 implies that every congruence subgroup is a ﬁnite union of polynomial families. There are congruence subgroups that are not polynomial families; see Proposition 5.13 and Corollary 5.14 below. Example 14. Consider the equation x2 x2 x2. Its integer solutions are 1 C 2 D 3 known as Pythagorean triples; sometimes the name is reserved for solutions that are primitive and/or positive. Let X be the set of all integer solutions 2 The equation can be written as x .x2 x3/.x3 x2/ so every element of 1 D C X gives a solution to the equation in Example 5. 986 LEONID VASERSTEIN

The set X is not a polynomial family but can be covered by two polynomial families: 2 2 2 2 2 2 2 2 .x1; x2; x3/ y3.2y1y2; y y ; y y / or y3.y y ; 2y1y2; y y /: D 1 2 1 C 2 1 2 1 C 2 The subset X 0 of all primitive solutions is the disjoint union of four families described by the same polynomials but with y3 1 and .y1; y2/ Um2 ޚ with D ˙ 2 odd y1 y2. C To get a polynomial parametrization of these pairs .y1; y2/ and hence to cover X 0 by four polynomials, we use Theorem 13. Let H be the subgroup of SL2 ޚ 0 1 generated by SL2.2ޚ/ and the matrix 1 0 . The ﬁrst rows of matrices in H are exactly the .a; b/ Um2 ޚ such that a b is odd. It follows from Theorem 13 2 C (see Example 5.12 below) that H is a polynomial family with 95 parameters. Thus, the set X 0 of primitive solutions is the union of four polynomial families with 95 parameters each. Example 15. Now we consider the equation x2 x2 x2 3. Finding 1 C 2 D 3 C its integer solutions was a famous open problem stated as a limerick a long time ago; it is[8, CNTA 5.14]. Using the obvious connection with the equation in our Example 7, Beukers[8] splits the set of solutions X into two families, each of them parametrized by the group H above (Example 14). So Theorem 13 implies that X is the union of two polynomial families, con- trary to the belief of Beukers[8]. Example 16. A few results of the last millennium,[5] and[3], together with our results, show that for arbitrary integers a; b; c and any integers ˛; ˇ; 1, the set of primitive solutions to the equation ax˛ bxˇ cx can be covered by a 1 C 2 D 3 ﬁnite (possibly, empty) set of polynomial families. Details will appear elsewhere. The minimal cardinality of the set is not always known; in the case of ˛ ˇ 3, D D the cardinality is 8 for even ˛ and 6 for odd ˛ (Fermat’s Last Theorem). In a future paper, using a generalization of Theorem 1 to rings of algebraic numbers, we will prove that many arithmetic groups are polynomial families. In Section 5 of this paper, we will consider only Chevalley-Demazure groups of clas- sical type, namely SLn ޚ, the symplectic groups Sp2n ޚ, the orthogonal groups SOn ޚ, and the corresponding spinor groups Spinn ޚ. Recall that

Sp ޚ is a subgroup of SL2n ޚ preserving the bilinear form 2n

x1y2 y1x2 x2n 1y2n y2n 1x2n; C C SO2n ޚ is a subgroup of SL2n ޚ preserving the quadratic form

x1x2 x2n 1x2n; C C POLYNOMIAL PARAMETRIZATION 987

SO2n 1 ޚ is a subgroup of SL2n 1 ޚ preserving the quadratic form C C 2 x1x2 x2n 1x2n x2n 1; C C C C there is a homomorphism (isogeny) Spin ޚ SOn ޚ whose kernel and cok- n ! ernel are both of order 2 (see[17]), Spin ޚ SL2 ޚ Sp ޚ (see Example 5), 3 D D 2 Spin ޚ SL2 ޚ SL2 ޚ (see Example 8), 4 D Spin ޚ Sp ޚ and Spin ޚ SL4 ޚ (see[21]). 5 D 4 6 D From Theorem 1, we easily obtain (see Section 4)

COROLLARY 17. For any n 2, (a) the group SLn ޚ is a polynomial family with 39 n.3n 1/=2 parameters, C C 2 (b) the group Spin2n 1 ޚ is a polynomial family with 4n 41 parameters, C C (c) the group Sp ޚ is a polynomial family with 3n2 2n 41 parameters, 2n C C (d) the group Spin ޚ is a polynomial family with 4.n 1/2 .n 1/ 36 2n 2 C C C parameters. C So SOn 1 ޚ is the union of two polynomial families. C The polynomial parametrization of SLn ޚ implies obviously that the group GLn ޚ is a union of two polynomial families for all n 1. (It is also obvious that GLn ޚ is not a polynomial family.) Less obvious is the following consequence of Corollary 17(a):

COROLLARY 18. For any integer n 1, the set Mn of all integer n n matri- 2 ces with nonzero determinant is a polynomial family in ޚn that has 2n2 6n 39 C C parameters.

Proof. When n 1, M1 is the set of nonzero integers. It is parametrized by D the polynomial 2 2 2 2 f .y1; y2; y3; y4; y5/ .y y y y 1/.2y5 1/ D 1 C 2 C 3 C 4 C C with ﬁve parameters. (We used Lagrange’s theorem asserting that the polynomial y2 y2 y2 y2 parametrizes all integers 0, but we did not use Corollary 17.) 1 C 2 C 3 C 4 Assume now that n 2. Every matrix ˛ Mn has the form ˛ ˇ, where 2 D ˇ SLn ޚ and is an upper triangular matrix with nonzero diagonal entries. Using 2 39 3n.n 1/=2 parameters for ˛ (see Corollary 17(a)), ﬁve parameters for each C C diagonal entry in (see the case n 1 above), and one parameter for each off- D diagonal entry in , we obtain a polynomial parametrization for Mn whose number of parameters is 39 3n.n 1/=2 5n n.n 1/=2 2n2 6n 39: C C C C D C C 988 LEONID VASERSTEIN

Remark 1. Similarly, every system of polynomial inequalities (with the in- equality signs , , , >, < instead of the equality sign in polynomial equations) ¤ Ä can be converted to a system of polynomial equations by introducing additional variables. For example, the set Mn in Corollary 18 is a projection of the set of all integer solutions to the .n2 5/-variable polynomial equation C 2 2 2 2 det.xi;j / .xn2 1 xn2 2 xn2 3 xn2 4 1/.2xn2 5 1/: D C C C C C C C C C C The polynomial parametrization of SLn ޚ with n 3 is related to a bounded generation of this group. In[4], it is proved that every matrix in SLn ޚ with n 3 is a product of 36 n.3n 1/=2 elementary matrices (for n 2, the number of C D elementary matrices is unbounded). Since there are n2 n types of elementary i;j matrices z with i j , this gives a polynomial parametrization of SLn ޚ for ¤ n 3, with .n2 n/.36 n.3n 1/=2/ C parameters. Conversely, any polynomial matrix

˛.y1; : : : ; yN / SLn.ޚŒy1; : : : ; xN / 2 is a product of elementary polynomial matrices[15] if n 3. When ˛.ޚN / D SLn ޚ, this gives a representation of every matrix in SLn ޚ as a product of a bounded number of elementary matrices. We conclude the introduction with remarks on possible generalization of The- orem 1 to commutative rings A with 1. If A is semi-local (which includes all fields and local rings) or, more generally, A satisfies the first Bass stable range condition sr.A/ 1 (which includes, e.g., the ring of all algebraic integers; see[19]), then D every matrix in SL2 A has the form Â1 u ÃÂ 1 0ÃÂ1 u ÃÂ 1 0Ã 1 3 ; 0 1 u2 1 0 1 u4 1 which gives a polynomial matrix

P.y1; y2; y3; y4/ SL2.ޚŒy1; y2; y3; y4/ 2 4 such that P.A / SL2 A. For any commutative A with 1, any N , and any poly- D nomial matrix

P.y1; : : : ; yN / SL2.ޚŒy1; : : : ; yN /; 2 n all matrices ˛ P.A / have the same Whitehead determinant wh.˛/ SK1 A; see 2 2 [1]. There are rings A, e.g.;A ޚŒp D for some D [2], such that wh.SL2 A/ D N D SK1 A 0. For such rings A, there in no N and P such that P.A / SL2 A. ¤ D Allowing coefﬁcients in A does not help much. For any matrix

P.y1; : : : ; yN / SL2.AŒy1; : : : ; yN /; 2 POLYNOMIAL PARAMETRIZATION 989

N all matrices in P.A / have the same image in SK1 A= Nill1 A, where Nill1 A is the subgroup of SK1 A consisting of wh.˛/ with unipotent matrices ˛. There are rings A such that wh.SL2 A/ SK1 A Nill1 A; see[2]. For such a ring A, there DN ¤ in no N and P such that P.A / SL2 A. D 1. Proof of Theorem 1

We denote elementary matrices by Â1 bÃ Â1 0Ã b1;2 and c2;1 : D 0 1 D c 1 1;2 2;1 It is clear that each of the subgroups ޚ and ޚ of SL2 ޚ is a polynomial family with one parameter. Note that the conjugates of all elementary matrices are covered by a polynomial matrix Â 2 Ã 1 y1y3y2 y1 y3 ˆ3.y1; y2; y3/ C 2 WD y y3 1 y1y3y2 2 a b in three variables. Namely, for ˛ SL2 ޚ, D c d 2 Â1 aec a2e Ã Â1 bed b2e Ã ˛e1;2˛ 1 and ˛e2;1˛ 1 C : D c2e 1 aec D d 2e 1 bed C 1;2 Remark 2. Conversely, every value of ˆ3 is a conjugate of b in SL2 ޚ for some b ޚ. 2 Next we denote by X4 the set of matrices of the form Âa bÃÂa c Ã Ä Â 0 1Ã Â 0 1Ã Â0 1Ã ˛˛T ˛; ˛ ˛ 1 ; D c d b d D 1 0 D 1 0 1 0 where ˛ SL2 ޚ. Since 2 Â 0 1Ã Â 1 0ÃÂ1 1ÃÂ 1 0Ã ; 1 0 D 1 1 0 1 1 1 we have Â1 bd b2 ÃÂ1 ac a2 ÃÂ1 bd b2 ÃÂ0 1Ã ˛˛T D d 2 1 bd c2 1 ac d 2 1 bd 1 0 C C C ˆ4.a; b; c; d/: DW Hence the set X4 is covered by a polynomial matrix ˆ4.y1; y2; y3; y4/ in four 4 variables: X4 ˆ4.ޚ /. 0 1 4 Remark 3. 1 0 ˆ4.0; 0; 0; 0/ ˆ4.ޚ /, while reduction modulo 2 shows D 0 1 2 that X4 does not contain 1 0 . 990 LEONID VASERSTEIN

Note that ˆ4. 1; 0; 0; 1/ diag.1; 1/. Therefore we can deﬁne the poly- ˙ ˙ D nomial matrix Â Ã Â Ã 1 y5 0 y5 0 ˆ5.y1; y2; y3; y4; y5/ ˆ4.1 y1y5; y2y5; y3y5; 1 y4y5/ D 0 1 C C 0 1

SL2.ޚŒy1; y2; y3; y4; y5/: 2 By deﬁnition,

1 Âe 0ÃÂ1 ae be ÃÂ1 ae ce ÃÂe 0Ã C C 0 1 ce 1 de be 1 de 0 1 C C Â 2 ÃÂ 2 Ã 1 ae be 1 ae ce 5 C C ˆ5.ޚ / D c 1 de b 1 de 2 C C if a; b; c; d; e ޚ, e 0, and the second matrix on the left side is in SL2 ޚ. 2 ¤ 5 For a; b; c; d; e ޚ, we denote by X5 ˆ5.ޚ / the set of matrices of the 2 form Â1 ae be2 ÃÂ1 ae ce2 Ã Â1 ae be2 Ã C C with C SL2 ޚ: c 1 de b 1 de c 1 de 2 C C C 1 0 The case e 0 is included because ˆ5.0; b; c; 0; 0/ b c 1 . D 1 T D C Note that X X5. Set Y5 X (the transpose of the set X5/. 5 D WD 5 Our next goal is to prove that every matrix in SL2 ޚ is a product of a small number of elementary matrices and matrices from X5 and Y5.

1 ae ce LEMMA 1.1. Let a; c; e ޚ, and let ˛ C SL2 ޚ. Then there are 2 D 2 u; v ޚ;" 1; 1 , and ' X5 such that the matrix 2 2 f g 2 1;2 2;1 1;2 1;2 2;1 ˛.eu/ v . c1e/ '. "ev/ . "u/ Â Ã has the form ; where c1 c u.1 ae/. "c 1 ae WD C C C Proof. The case 1 ae 0 is trivial (we can take u v 0 and " e/, C D D D D so we assume that 1 ae 0. By Dirichlet’s theorem on primes in arithmetic C ¤ progressions, we ﬁnd u ޚ such that either c1 c u.1 ae/ 1 mod 4 2 WD C C Á and c1 is a prime or c1 c u.1 ae/ 3 mod 4 and c1 is a prime. Then WD C C Á GL1.ޚ=c1ޚ/ is a cyclic group, and the image of 1 in this group is not a square. 2 2 So a a mod c1 for some a1 ޚ. We write a vc1 "a with v ޚ and D ˙ 1 2 C D 1 2 " GL1 ޚ. Then, for some b1; d1 ޚ, 2 2 Â 2 Ã Â 2 2 2Ã 1;2 2;1 1;2 1 "a1e c1e 1;2 1 "a1e "c1e a1 ˛.ue/ v . c1e/ C . c1e/ C : D D b1 d1 POLYNOMIAL PARAMETRIZATION 991

Let ˇ be the rightmost matrix above, and note that

Â 2 2 Ã 1 d 1 "c1a1e ˇ 2 : D b1 1 "a e C 1 Since det.ˇ/ 1, we conclude that d1 1 eޚ. D 2 We set Â 2 2Ã Â Ã d1 b1a1e 2 : WD "c1 1 "a e D "c1 1 .a vc1/e C 1 C C 1 Then ' ˇ X5 and ˇ'. WD 2 D Now Â Ã . "ev/1;2. "u/2;1 . "u/2;1 D "c1 1 ae C Â Ã Â Ã . "u/2;1 : D ".c u.1 ae// 1 ae D "c 1 ae C C C C a b LEMMA 1.2. Let ˛ SL2 ޚ, with m 1 an integer. Then there are D 2 zi ޚ, ' X5, and Y5 such that 2 2 2 Âam bÃ ˛mz1;2z2;1z1;2'z1;2z2;1z1;2z2;1 z2;1z1;2z2;1 : 1 2 3 4 5 6 7 8 9 10 D Proof. By the Cayley-Hamilton theorem and mathematical induction on m, we have Â Ã m f ga gb ˛ f 12 g˛ C with f; g ޚ, D C D 2 m 2 where 12 is the identity matrix. Since 1 = det(˛ / f mod g, we can write Á g g1g2 with f 1 mod g1 and f 1 mod g2. D Á Á By Lemma 1.1, there are z1; z2; z3; z4; k1 ޚ and '1 X5 such that 2 2 Â Ã m 1;2 2;1 1;2 1;2 2;1 ˛ z1 z2 z3 '1z4 k1 ˇ ˇ : DW D D g2b f ga ˙ C Now we apply Lemma 1.1 to the matrix Â Ã Â Ã 1 Â Ã 0 1 0 1 f ga g2b ˇ ˙ D 1 0 1 0 D instead of ˛. So there are k2; z6; z7; z8; z9 ޚ and ' X5 such that 2 0 2 Â Ã 1;2 2;1 1;2 1;2 2;1 k . z6/ . z7/ ' . z8/ . z9/ : 2 0 D b f ga ˙ 992 LEONID VASERSTEIN

1 1 Negating this and conjugating by 0 1 we obtain that 0 Â Ã 2;1 1;2 2;1 2;1 1;2 f ga b ˇ. k2/ z z z z C ˙ ; 6 7 8 9 D D 0 1 1 0 1 where ' Y5. WD 1 0 0 1 0 2 The matrix ˛ is lower triangular modulo b, so f ga am mod b. We m C Á find z10 ޚ such that f ga z10b a and set z5 k1 k2 to obtain our 2 C ˙ D D conclusion. a b COROLLARY 1.3. Let ˛ SL2 ޚ, with m 1 an integer, and " 1 . m D 2 2 f˙ g Assume that a " mod b. Then there are zi ޚ and 'i X5 such that Á 2 2 m 1;2 2;1 1;2 1;2 2;1 1;2 2;1 2;1 1;2 2;1 1;2 2;1 ˛ z z z '1z z z z '2z z z z z "12: 1 2 3 4 5 6 7 8 9 10 11 12 D Proof. By Lemma 1.2, we find t1; zi ޚ with 1 i 9, ' X5, and Y5 2 Ä Ä 2 2 such that the matrix Âam bÃ ˇ ˛mz1;2z2;1z1;2'z1;2z2;1z1;2z2;1 z2;1z1;2t 2;1 : WD 1 2 3 4 5 6 7 8 9 1 D 2;1 1;2 2;1 Now we can find t2; z11; z12 ޚ such that ˇt z z "12. The conclusion 2 2 11 12 D follows from setting z10 t1 t2. D C For any integer s 1, we denote by the s the following polynomial matrix in s parameters: 1;2 2;1 s.y1; : : : ; ys/ y y ; D 1 2 1;2 2;1 where the last factor is the elementary matrix ys (resp., ys / when s is odd (resp., even). We set 1 Â 0 1Ã Â 0 1Ã s.y1; : : : ; ys/ s.y1; : : : ; ys/ : D 1 0 1 0 Note that 1 T s.y1; : : : ; ys/ s.y1; : : : ; ys/ s.y1; : : : ; ys/ for even s, D D 1 T s.y1; : : : ; ys/ s.y1; : : : ; ys/ s.y1; : : : ; ys/ for odd s, D D where “T ” denotes transpose, and that s.y1; : : : ; ys/ s 1.0; y1; : : : ; ys/ for D C any s. a b COROLLARY 1.4. Let ˛ SL2 ޚ and " 1 . Assume that for some D c d 2 2 f˙ g coprime integers m; n 1, we have am 1 mod b and an 1 mod c. Then i Á ˙i Á ˙ there are " 1 , ıi i .ޚ /, i i .ޚ /, 'i X5, and i Y5 such that 2 f˙ g 2 2 2 2 ˛ " 5'1 4 2ı7 1ı4'2 3: D POLYNOMIAL PARAMETRIZATION 993

Proof. Replacing m and n by their positive multiples, we can assume that n m 1. By Corollary 1.3, D m ˛ 5'1 4'2ı3 D ˙ 3 4 5 with '1;'2 X5, ı3 3.ޚ /, 4 4.ޚ /, and 5 5.ޚ /. 2 2 2 2 Applying Corollary 1.3 to ˛T instead of ˛, we get .˛T /n ' ' ı D ˙ 50 10 40 20 30 3 i with 'i0 X5, ı30 3.ޚ /, and i0 i .ޚ /. 2 2 0 1 2 Conjugating by 1 0 , we obtain that Â Ã Â Ã 1 n 0 1 T n 0 1 ˛ .˛ / ı5 1ı4 2 3 D 1 0 1 0 D ˙ i with i Y5, 3 3, and ıi 4.ޚ /. 2 2 2 Therefore m n 7 ˛ ˛ ˛ 5'1 4'2ı7 1ı4 2 3; where ı7 ı3ı5 7.ޚ /. D D ˙ WD 2 PROPOSITION 1.5. Every matrix ˛ SL2 ޚ can be represented as 2 ˛ 5'1 4'2ı7 1ı4 2 6 D i i with ıi i .ޚ /, i i .ޚ /, 'i X5, and i Y5. 2 2 2 2 Proof. Let ˛ a b . The case a 0 is trivial, so let a 0. As in the proof D c d D ¤ of Lemma 1.1, can ﬁnd an integer u such that b au is a positive prime 3 j C j Á modulo 4. Then we ﬁnd an integer v such that c av is a positive prime such that C GCD.c av 1; b au 1/ 1 or 2. C j C j D Let now m . b au 1/=2 and n c av 1. Then GCD.m; n/ =1, i.e., D j C j D C m m; n are coprime, i.e., .m; n/ Um2 ޚ. Moreover a 1 mod .b au/ and 2 Á ˙ C am 1 mod .c av/. Á C By Corollary 1.4, we have 2;1 1;2 v ˛u '1 4'2ı7 1ı4 2 3 D ˙ 50 i i with ıi i .ޚ /, i ; i0 i .ޚ /, 'i X5, and i Y5. 2 2;1 2 5 2 21;2 4 Set 5 . v/ 50 5.ޚ / and 40 3. u/ 4.ޚ /. It remains to D 42 4 D i 2 i 2 observe that 12 4.ޚ / 4.ޚ /; hence i .ޚ / i 2.ޚ C / for all i 1. ˙ 2 \ ˙6 C In particular, both and belong to 6.ޚ /. 40 40 Note that Theorem 1 follows from Proposition 1.5. The polynomial parametrization of SL2 ޚ in Proposition 1.5 is explicit enough to see that the number of parameters is 46 and the total degree is at most 78. This is because the degrees of s and s are both s and the degree of ˆ5 is 13. 994 LEONID VASERSTEIN

2. Primitive vectors and systems of linear equations

First, Proposition 1.5 yields a lemma:

i LEMMA 2.1. For any .a; b/ Um2 ޚ there are ıi ; ıi0 i .ޚ /, 'i X5, 4 2 2 2 4; 6 i .ޚ /, and i Y5 such that 2 2 .a; b/ .1; 0/ı '1 4 2ı7 1ı4'2 6: D 40 Proof. Let ˛ be a matrix in SL2 ޚ with the ﬁrst row .a; b/. We write ˛ as in Proposition 1.5. Multiplying by the row .1; 0/ on the left, we obtain

.a; b/ .1; 0/˛ .1; 0/ 5'1 4 2ı7 1ı4'2 6: D D Since

.1; 0/s.y1; : : : ; ys/ .1; 0/s 1.y2; : : : ; ys/; D 4 we can replace 5 by ı 4.ޚ /. 40 2 The lemma implies the following result:

THEOREM 2.2. The set Um2 ޚ of coprime pairs of integers admits a polynomial parametrization with 45 parameters.

For n 3, it is easy to show that Umn ޚ admits a polynomial parametrization with 2n parameters. This is because the ring ޚ satisﬁes the second Bass stable range condition. Now we introduce this condition. n A row .a1; : : : ; an/ A over an associative ring A with 1 is said to be 2 P unimodular if a1A anA A, i.e., there are bi A such that ai bi 1. C C D n2 D Let Umn A denote the set of all unimodular rows in A . We write sr.A/ n if for any .a1; : : : ; an 1/ Umn 1 A there are ci A Ä C 2 C 2 such that .a1 an 1c1; : : : ; an an 1cn/ Umn A. C C C C 2 For example, it is easy to see that sr.A/ 1 for any semi-local ring A and that Ä sr(ޚ/ 2. Ä It is shown in[16] that for any m the condition sr.A/ m implies that Ä sr.A/ n for every n m. Moreover, if sr.A/ m and n m 1, then for any Ä Ä C a .a1; : : : ; an/ Umn A there are c1; : : : ; cm A such that a0 .ai0 / Umn 1 A, D 2 2 D 2 where ai0 ai anci for i 1; : : : ; m and ai0 ai for i m 1; : : : ; n 1. D C D P D D C Using now bi A such that a bi 1, we obtain that 2 i0 D m n 1 n 1 Y n;i Y i;n Y n;i a c .bi .1 an// . a / .0; : : : ; 0; 1/: i i0 D i 1 i 1 i 1 D D D Here xi;j denotes an elementary matrix with x at position .i; j /. We denote by EnA the subgroup of GLn A generated by these elementary matrices. POLYNOMIAL PARAMETRIZATION 995

Thus, there is a polynomial matrix ˛ En.ޚ y1; : : : ; y2n m 2 / (with non- 2 h C i commuting yi ) that is a product of 2n m 2 elementary matrices, such that C 2n m 2 Umn A is the set of last rows of all matrices in ˛.ޚ C /. In particular, taking A ޚ and m 2, we obtain this: D D PROPOSITION 2.3. For any n 3, the set Umn ޚ is a polynomial family with 2n parameters. Now we are ready to prove Corollary 2. Consider now an arbitrary system x b of l linear equations for k variables x with integer coefficients. We write x D and solutions as rows. Reducing the coefficient matrix to a diagonal form ˛ ˇ (where ˛ SL ޚ, ˇ SL ޚ, and the diagonal entries are the Smith invariants) by 2 k 2 l row and column addition operations with integer coefficient, we write our answer, describing all integer solutions, in one of these three forms: (1) 0 1 (there are no solutions); D (2) x c˛, where c ޚk (so c˛ is only solution); D 2 (3) x y˛, where y is the row of k parameters (i.e., x is arbitrary); D (4) x .a; y/˛ with a row y of N parameters (1 N k 1) and a ޚk N . D Ä Ä 2 Thus, the set X of all solutions, when it is not empty, is a polynomial family with N parameters (0 N k) and the degree of parametrization is at most 1. Ä Ä Now we are interested in the set Y primitive solutions. In case (1), Y is empty. In case (2), Y either is empty or consists of a single solution. In case (3), N k and Y UmN ޚ, which is a polynomial family by D D Theorem 2.2 and Proposition 2.3 provided that N 2. When N 1, we have D ˛ 1, and the set Y Um1 ޚ 1 is not a polynomial family, but consists D ˙ D D f˙ g of two constant polynomial families. In case (4) with a 0 (the homogeneous case), we have N < k and the set D Y is also parametrized by UmN ޚ. Now we have to deal with case (4) with a 0. Let d GCD.a/. Then ¤ D Y is parametrized by the set Z b ޚN GCD.d, GCD.b// 1 . We find a f D f 2 W D g polynomial f .t/ ޚŒt whose range reduced modulo d is GL1.ޚ=dޚ/. (Find f .t/ 2 modulo every prime p dividing d and then use the Chinese Remainder Theorem; the degree of f .t/ is at most the largest p 1.) Then the range of the polynomial f2.t1; t2/ f .t1/ dt2 consists of all WD C integers z such that GCD.d; z/ 1. Therefore the set Z consists of f2.z1; z2/u D with z1; z2 ޚ and u UmN ޚ. Thus, any polynomial parametrization of UmN ޚ 2 2 yields a polynomial parametrization of Z (and hence Y ) with two additional parameters. By Theorem 2.2 and Proposition 2.3, the number of parameters is at most 41 2k (at most 2k when N 3/. C 996 LEONID VASERSTEIN

3. Quadratic equations

In this section we prove Corollary 4, which includes Corollary 3 as a particular case with Q0 1, k 2n, and D D Q0.x5; : : : ; xk/ x5x6 x2n 1x2n D C C k .Q 0 when n 2/. We write the k-tuples in ޚ as rows. Let e1; : : : ; e be the 0 D D k standard basis in ޚk. We denote by SO.Q; ޚ/ the subgroup of SLn ޚ consisting of matrices a 2 SLn ޚ such that Q.x˛/ Q.x/. In the end of this section, we prove that, under the D conditions of Corollary 4, the group SO.Q; ޚ/ consists of two disjoint polynomial families. k We deﬁne a bilinear form . ; /Q on ޚ by .a; b/Q Q.a b/ Q.a/ Q.b/. D C Following[20], we introduce elementary transformations .e; u/ SO.Q; ޚ/; 2 where e e1 or e2 and .e; u/Q 0, by setting D D v.e; u/ v .e; v/Qu .u; v/Qe Q.u/.e; v/Qe D C (this works because Q.e/ 0/. D LEMMA 3.1. Let Q be as in Corollary 4. Then for any Q-unimodular row k Pk k v ޚ , there are u; u0 U i 5 ޚei ޚ such that the ﬁrst four entries of the 2 2 D D row v.e1; u/.e2; u0/ form a primitive row.

Proof. We write v .v1; v2; : : : ; v /. First we want to ﬁnd a u U such that D k 2 ޚv ޚv3 ޚv4 0, where v .v / v.e1; u/ (note that v vi for i 2; 3; 4/. 10 C C ¤ 0 D i0 D i0 D D If ޚv1 ޚv3 ޚv4 0, we can take u 0. C C ¤ D Otherwise, since v is Q-unimodular, ޚv2 ޚ.v; w/Q 0 for some w U . C ¤ 2 2 For v .v / v.e1; cw/ with c ޚ, we have v v1 .v; w/Qc Q.w/v2c 0 D i0 D 2 10 D is a nonconstant polynomial in c (with v1 0/, so it takes a nonzero value for D some c. Therefore we can set u cw with this c. D Now we want to ﬁnd u U such that 0 2 .v ; v ; v ; v / .v ; v ; v3; v / Um4 ޚ; 100 200 300 400 D 10 200 40 2 where w .v / v .e2; u / v.e1; u/.e2; u /: 00 D i00 D 0 0 D 0 Since v is Q-unimodular, there is a w U such that 0 0 2 .v ; v2; v3; v4; .v ; w /Q/ Um5 ޚ: 10 0 0 2 Since ޚv ޚv3 ޚv4 0, there is a c ޚ such that 10 C C ¤ 0 2 .v ; v2 c .v ; w /Q; v3; v4/ Um4 ޚ: 10 0 0 0 2 POLYNOMIAL PARAMETRIZATION 997

We set u c w . Then v .e2; u / .v / with 0 D 0 0 0 0 D i00 2 .v ; v ; v ; v / .v ; v2 c .v ; w /Q c Q.w /v ; v3; v4/ Um4 ޚ: 100 200 300 400 D 10 0 0 0 0 0 10 2 LEMMA 3.2. Let k 4, Q0 ޚ, Q any quadratic form in k 4 variables, 2 0 and Q.x1; : : : ; x / x1x2 x3x4 Q .x5; : : : ; x /. Then the set X of integer so- k D C C 0 k 0 lutions for the equation Q.x/ Q0 with .x1; x2; x3; x4/ Um4 ޚ is a polynomial D 2 family with k 88 parameters. C Proof. When k 4, see Examples 6 and 8. Assume now that k 5. Let D v .vi / X . Set D v1v2 v3v4 ޚ. We can write D 2 0 D C 2 Â Ã Â Ã v1 v3 1 1 0 ˛ ˇ v4 v2 D 0 D with ˛; ˇ SL2 ޚ. Then we can write 2 Pk .1; D; 0; 0; v5; : : : ; vk/ .1; Q0; 0; : : : ; 0/.e2; i 5 vi ei /: D D Therefore X is parametrized by k 5 parameters v5; : : : ; v and two matrices in k SL2 ޚ. By Theorem 1;X is a polynomial family with k 4 2 46 k 88 C D C parameters. Combining Lemmas 3.1 and 3.2, we obtain Corollary 4. LEMMA 3.3. Under the conditions of Lemma 3.2, assume that k 6 and that Q .x5; : : : ; x / x5x6 Q .x7; : : : ; x /. Then the set X is a polynomial family 0 k D C 00 k 0 with k 2 parameters. C Proof. Let .vi / X . There is an orthogonal transformation ˛ SO4 ޚ 2 0 2 (coming from Spin ޚ SL2 ޚ SL2 ޚ; see Example 8) such that 4 D .v1; v2; v3; v4/˛ .1; v1v2 v3v4; 0; 0/: D C 1 We set .w1; w2; w3; w4/ .0; 1; 0; 0/˛ and D k w e1w1 e2w2 e3w3 e4w4 ޚ : D C C C 2 Then Q.w/ 0 .w; v/Q. D D Consider the row v .v / v.v5; .1 v5/w/. For i 1; 2; 3; 4, we have 0 D i0 D D vi0 vi .1 v5/wi . Also v50 1, and vi0 vi for i 6. D C P D D So v .e6; v / e5; hence X is parametrized by 4 .k 2/ k 2 0 i 5;6 i0 D 0 C D C parameters. ¤ Combining Lemmas 3.1 and 3.3, we obtain:

PROPOSITION 3.4. Let k 6, Q0 ޚ, Q a quadratic form in k 6 variables, 2 00 and Q.x1; : : : ; x / x1x2 x3x4 x5x6 Q .x7; : : : ; x /. Then the set of all k D C C C 00 k Q-unimodular solutions for the equation Q.x/ Q0 is a polynomial family with D 3k 6 parameters. 998 LEONID VASERSTEIN

4. Chevalley-Demazure groups

We prove here Corollary 17. Let n 2, and take e1; : : : ; en as the standard basis of ޚn. First we prove by induction on n that SLn ޚ admits a polynomial factorization with 39 n.3n 1/=2 parameters. The case n 2 is covered by Theorem 1. Let C C D n 3. We consider the orbit en SL.n; ޚ/. The orbit admits a parametrization by 2n parameters by Proposition 2.3. More- over, there is a polynomial matrix ˛ En.ޚŒy1; : : : ; y2n/ that is a product of 2n 2 2n elementary matrices, such that Umn ޚ en˛.ޚ /. D The stationary group of en consists of all matrices of the form Âˇ vÃ ; where vT ޚn 1. 0 1 2 By the induction hypothesis, the stationary group can be parametrized by 39 C .n 1/.3n 2/=2 n 1 parameters. Thus SLn ޚ can be parametrized by 39 C C .n 1/.3n 2/=2 n 1 2n 39 n.3n 1/=2 parameters. C C D C C Now we consider the symplectic groups Sp2n ޚ. We prove Corollary 17(c) by induction on n. When n 1, Sp ޚ SL2 ޚ. Assume now that n 2. D 2 D As in[2], using that sr .ޚ/ 2, we have a matrix D ˛ Sp .ޚŒy1; : : : ; y4n/ such that e2n˛ Um2n ޚ. 2 2n D The stationary group consists of all matrices of the form 0 ˇ 0 v1 T T 2n 2 @v 1 cA ; where v ޚ and c ޚ, 2 2 0 0 1 so by the induction hypothesis it is parametrized by 2.n 1/2 2.n 1/ 41 2n 1 C C C parameters. Therefore Sp2n ޚ is parametrized by 2.n 1/2 2.n 1/ 39 2n 1 4n 3n2 2n 41 C C C C D C C parameters. Now we discuss polynomial parametrizations of the spinor groups

Spin ޚ Spin.Q2n; ޚ/ for n 3: 2n D We prove Corollary 17(d) by induction on n. When n 3, Spin ޚ SL4 ޚ. D 2n D Namely SL4 ޚ acts on alternating 4 4 integer matrices preserving the pfafﬁan, which is a quadratic form of type x1x2 x3x4 x5x6; see e.g.[21]. C C POLYNOMIAL PARAMETRIZATION 999

2n Assume now that n 4. The group Spin ޚ acts on ޚ via SO2n ޚ. The 2n orbit e2n SO2n ޚ of e2n is the set of all unimodular ( Q2n-unimodular) solutions D for the equation Q2n 0. By Proposition 3.4, the orbit is parametrized by 6n 6 D parameters. Also, the matrices . ; / come from Spin ޚ, so there is a polyno- 2n mial matrix in Spin ޚ with 6n 6 parameters that parametrizes the orbit. The 2n stationary subgroup in SO2n ޚ consists of the matrices of the form 0 ˇ 0 v1 t T 2n 2 T @v 1 cA ; where v ޚ , c Q2n 2.v / ޚ, ˇ SO2n 2 ޚ: 2 D 2 2 0 0 1

By the induction hypothesis, the stationary subgroup of e1 in Spin2n ޚ is parametrized by 4.n 1/2 .n 1/ 34 2n 2 C C parameters. So Spin2n ޚ is a polynomial family with 4.n 1/2 .n 1/ 36 2n 2 6n 3 4n2 n 36 C C C D C parameters. Finally, we prove Corollary 17(b) by induction on n. When n 2, Spin ޚ D 5 D Sp ޚ (the group Sp ޚ SL4 ޚ acts on the alternating matrices as above, ﬁxing a 4 4 vector of length 1) and the formula works. Let now n 3. The orbit e1 SO2n 1 ޚ of e1 is the set of all unimodular C ( Q2n-unimodular) solutions of the equation Q2n 1 0. By Proposition 3.4, D C D the orbit is parametrized by 3.2n 1/ 6 6n 3 parameters. Moreover, the C D matrices . ; / come from Spin ޚ, so there is a polynomial matrix in Spin ޚ 2n 2n with 6n 3 parameters that parametrizes the orbit. The stationary subgroup of e1 in SO2n 1 ޚ consists of the matrices of the form C 0 1 0 01 2n 1 @ c 1 vA ; where v ޚ , c Q2n 1.v/ ޚ, ˇ SO2n 1 ޚ: 2 D 2 2 vT 0 ˇ

By the induction hypothesis, the stationary subgroup of e1 in Spin2n 1 ޚ is parametrized by 4.n 1/2 41 2n 1 parameters. So Spin ޚ is a polynomial C C 2n family with 4.n 1/2 41 2n 1 6n 3 4n2 41 C C C D C parameters.

Remark 4. As in Corollary 18, for any square-free integer D or D 0, we D obtain a polynomial parametrization of the set of all integer n by n matrices with determinant D. If D is not square-free, the set of matrices is a ﬁnite union of polynomial families. 1000 LEONID VASERSTEIN

5. Congruence subgroups

In this section we fix an integer q 2. Denote by G.q/ the subgroup of SL2 ޚ a b 2 consisting of matrices c d such that b; c qޚ and a 1; d 1 q ޚ. This group 2 2 2 is denoted by G.qޚ; qޚ/ in[18]. Note that SL2.q ޚ/ G.q/ G. q/ SL2 qޚ. 2 D We parametrize G.q/ by the solutions of x1 x4 q x1x4 x2x3 0 as C C D Â 2 Ã 1 q x1 qx2 x1; x2; x3; x4 C 2 . 7! qx3 1 q x4 C We use the polynomial matrices ˆ4.y1; y2; y3; y4/ and ˆ5.y1; y2; y3; y4; y5/ 2 2 defined in Section 1. We denote by X4.q/ ˆ4.1 q ޚ; qޚ; qޚ; 1 q ޚ/ G.q/ T C C T the set of matrices of the form ˛˛ with ˛ G.q/. Notice that X4.q/ 1 2 D X4.q/ X4.q/. D 2 2 We denote by X5.q/ ˆ5.q ޚ; qޚ; qޚ; q ޚ; ޚ/ G.q/ the set of matrices of the form Â1 aq2e bqe2 ÃÂ1 aq2e cq2e2 Ã C C cq 1 dq2e bq 1 dq2e C C with Â1 aq2e bqe2 Ã a; b; c; d; e ޚ and C SL2 ޚ: 2 cq 1 dq2e 2 C Set Â Ã 1 Â Ã 1 T 0 1 0 1 Y5.q/ .X5.q/ / X5.q/ : D D 1 0 1 0 T 1 T 1 Notice that X5.q/ Y5.q/ Y5.q/ and Y5.q/ X5.q/ X5.q/ D D D D We also use the polynomial matrices i ; i defined in Section 1. Notice that i i i .qޚ /; i .qޚ / G.q/ and that 2i T 2i 2i 1 2i 2i .qޚ / 2i .qޚ /; 2i .qޚ / 2i .qޚ /; D D 2i 1 T 2i 1 2i 1 1 2i 1 2i 1.qޚ / 2i 1.qޚ /; 2i 1.qޚ / 2i 1.qޚ / D D for all integers i 1. LEMMA 5.1. Let Â1 aq2e cqeÃ a; c; e ޚ; e 0; ˛ C G.q/: 2 ¤ D 2 i Then there are ıi i .qޚ /, " 1; 1 , and ' X5.q/ such that 2 2 f g 2 Â Ã ˛ı3'ı2 : D "cq 1 aq2e C POLYNOMIAL PARAMETRIZATION 1001

Proof. As in the proof of Lemma 1.1 above, we ﬁnd u; v ޚ such that 2 2 2 2 c u.1 aq e/ is a prime 3 modulo 4 and a vq c1 "a , where j C C j Á C D 1 2 c1 c u.1 aq e/; a1 ޚ;" GL1 ޚ: WD C C 2 2 1;2 2;1 1;2 3 Set ı3 .uqe/ .vq/ . c1eq/ 3.qޚ /. Then, for some b1; d1 ޚ, D 2 2 Â 2 2 Ã Â 2 2 2 3 2Ã 1 "a1q e c1qe 1;2 1 "a1q e "c1e q a1 ˛ı3 C . c1eq/ C D D b1 d1 ˇ G.q/: DW 2 Note that Â 2 3 2 Ã 1 d 1 "c1a1q e ˇ 2 2 : D b1 1 "a q e C 1 Set Â 2 2 2Ã Â Ã d1 b1a1e q 2 2 2 : WD "c1q 1 "a q e D "c1q 1 .a vc1/q e C 1 C C 1 Then ' ˇ X5.q/ and ˇ'. WD 2 D Now Â Ã 1;2 2;1 2;1 . "evq/ . "uq/ 2 . "uq/ D "c1q 1 aq e C Â Ã Â Ã . "u/2;1 ; D ".c u.1 aeq2//q 1 ae D "cq 1 aq2e C C C C 1;2 2;1 so we can take ı2 . "evq/ . "uq/ X2.q/. WD 2 LEMMA 5.2 (reciprocity). Let a; b ޚ and 2 Â1 aq2 .1 bq2/qÃ ˛ C C G.q/: D 2 Then there are ';' X5.q/ such that 0 2 Â1 bq2 .1 aq2/qÃ q1;2˛. q/1;2'. q/1;2' . q/1;2 C C : 0 D Proof. We have Â1 aq2 .b a/q3Ã ˛ ˛. q/1;2 C G.q/: 0 D D c d 2

Â 2 2Ã 1 1 1 aq cq Set ' ˛ C X5.q/ WD 0 .b a/q d 2 I 1 Â1 aq2 cq2Ã Â d cq2 Ã hence ˛ ˛ ' C : 00 D 0 D .b a/q d D .b a/q 1 aq2 C 1002 LEONID VASERSTEIN

Â d c q2 Ã Now q1;2˛ . q/1;2 0 0 : 00 D .b a/q 1 bq2 C Â 2 Ã 1 Â 3Ã 1 d 0 c0q d 0 .b a/q Set ' X5.q/ 0 WD .b a/q 1 bq2 c 1 bq2 2 I C 0 C hence 1 Âd .b a/q3Ã Â1 bq2 .b a/q3Ã ˇ q1;2˛ . q/1;2' 0 C : WD 00 0 D c 1 bq2 D c d 0 C 0 0 Â1 bq2 .1 aq2/qÃ Finally, ˇ. q/1;2 C C : D a b LEMMA 5.3. Let ˛ G.q/ . Then there are D c d 2 i X4.q/; ıi i .qޚ /; 1 1qޚ; ; Y5.q/; " 1 2 2 2 0 2 D ˙ such that Â Ã 2;1 2 2;1 2;1 . q/ ˛ ı3'ı2q q 1 : 0 D "b2 a T 1 Proof. Set .˛ ˛/ X4.q/. Then D 2 Â1 b.c b/ a.b c/ Ã ˛2 ˛.˛ 1/T C : D D d.c b/ 1 c.b c/ i By Lemma 5.1 with e .b c/=q, there are ıi i .qޚ /, " 1; 1 , and D 2 2 f g ' X5.q/ such that 2 Â Ã ˛ ı3'ı2 ˇ: D "aq 1 b.c b/ DW C Now we apply Lemma 5.2 to the matrix Â1 b.c b/ "aqÃ .ˇ 1/T C D and ﬁnd ; Y5.q/ such that 0 2 Â Ã . q/2;1ˇq2;1q2;1 q2;1 : D 0 D ".1 b.c b//q a C Since 1 b.c b/ ad b2, we have C D Â Ã 2;1 ."dq/ 1 : D D "b2 a LEMMA 5.4. Let Â1 aq2e cqeÃ a; c; e ޚ; e 0; ˛ C G.q/; " 1 : 2 ¤ D 2 0 2 f˙ g POLYNOMIAL PARAMETRIZATION 1003

i Then there are ıi i .qޚ / and ' X5.q/ such that 2 2 Â Ã ˛ı5'ı2 : D " cq 1 aq2e 0 C Proof. We ﬁnd u; v as in the proof of Lemma 5.1. Now we ﬁnd w ޚ such 2 2 2 that c2 is a prime 1 mod 4, where c2 c1 .1 "a1q e/w. Then there are j j Á 2 2 WD C C z; a2 ޚ such that "a zc2 " a . We set 2 1 C D 0 2 1;2 2;1 1;2 2;1 1;2 3 ı5 .uqe/ .vq/ .wqe/ .zq/ . c2eq/ 5.qޚ /: WD 2 Then Â 2 2 Ã Â 2 2 2 3 2Ã 1 "0a2q e c2qe 1;2 1 "0a2q e "0c2e q a1 ˛ı5 C . c3eq/ C G.q/: D D b1 d1 2 The rest of our proof is the same as that of Lemma 5.1.

a b i 3 LEMMA 5.5. Let ˛ G.q/. Then there are ıi i .qޚ /, 3 3.qޚ /, D c d 2 2 2 ' X5.q/, X4.q/, and Y5.q/ such that 2 2 2 Â 2 Ã 2 a b ı1˛ ı5'ı4 3 ˙ : D Proof. By Lemma 5.4 with e " 1, we ﬁnd D D 5 2 ı5'ı2 5.qޚ /X5.q/2.qޚ / D 2 such that Â Ã Âd c Ã ˛ 0 0 : D b a D b a 1 T 1 Set .˛ / ˛ X4.q/. Then D 2 Â d cÃ ˛ .˛ 1/T : D D b a Set Â ÃÂ Ã 1 d 0 c0 d c ı1 1.qޚ/: D b a b a 2 Then Â ÃÂ Ã Â Ã 2 d 0 c0 d 0 c0 ı1˛ D b a b a D b.a d / a2 bc 0 0 Â d c Ã 00 00 ˇ G.q/ D b.a d / 1 a.a d / DW 2 0 C 0 because ad bc 1. 0 0 D 1004 LEONID VASERSTEIN

By Lemma 5.1, Â Ã 1 T .ˇ / ı3' ı 0 20 D b 1 a.a d / ˙ C 0 i with ıi ; ı i .qޚ / and ' X5.q/; hence i0 2 0 2 Â ÃÂ 2 Ã 1 a.a d 0/ b a bc0 b ˇ 2 C ˙ ˙ 30 D i with i ; i0 i .qޚ / and Y5.q/. 2 2 4 2;1 3 Finally, we set ı4 ı2 4.qޚ / and 3 2. c / 3.qޚ / to D 30 2 D ˙ 0 2 obtain the conclusion.

a b i LEMMA 5.6. Let ˛ G.q/. Then there are ıi i .qޚ /, 1 1.qޚ/, D c d 2 2 2 ';' X5.q/, X4.q/, and Y5.q/ such that 0 2 2 2 Â Ã 1;2 2 1;2 . q/ ˛ ı3'ı2 q ' 1 : 0 D b2q a ˙ 1 1 T Proof. Set ˛ .˛ / X4.q/, so D 2 Â d cÃ Â d cÃ Â1 b.b c/ a.b c/Ã ˛ .˛ 1/T ; ˛2 ˛ : D D b a D b a D i By Lemma 5.1 with e .b c/=q, there are ıi i .qޚ / and ' X5.q/ D 2 0 2 such that Â Ã 2 ˛ ı3'ı2 ˇ: D .1 b.b c//q a DW ˙ T 1 1 b.b c/ aq Now we apply Lemma 5.2 to the matrix .ˇ / ˙ instead D of ˛. So Â Ã 1;2 T 1 1;2 1;2 a .1 b.b c//q q .ˇ / '. q/ ' . q/ ˙ ; with ';' X5.q/; 0 D 0 2 hence Â Ã 2;1 1;2 2;1 2;1 . q/ ˇq q q ˇ ; with ; Y5.q/. 0 D .1 b.b c//q a DW 0 0 2 ˙ Since .1 b.b c// ad b2, we have D Â Ã 2;1 ˇ for . dq/ 1.qޚ/. 0 10 D b2q a 10 D 2 ˙ 2;1 2;1 Finally, we set 1 q .ޚ/; ı2 ı q . WD 10 2 D 20 POLYNOMIAL PARAMETRIZATION 1005

COROLLARY 5.7. Let ˇ G.q/. Then there are 2 i ıi i .qޚ /; '; ' X5.q/; Y5.q/; 2 0 2 2 i a bq i i .qޚ /; X4.q/; ˛ G.q/ 2 2 D cq d 2 1;2 2 such that ı2ˇı . q/ ' 2' ı3 ˛ , where b and c are positive odd primes 20 0 D j j j j not dividing q, and GCD. b 1; c 1/ 2. j j j j D a0 b0q Proof. Let ˇ c q . The case c0 0 is trivial, so we assume that c0 0. D 0 d 0 D ¤ We ﬁnd u; v; b ޚ such that a d c uq2 is an odd prime and b2q2 c av. 2 WD 0 C 0 ˙ D 0 C Replacing, if necessary, b by b wa, we can assume that b is a positive odd prime C not dividing q. Then Â Ã ˇ ˇ.uq/1;2.vq/2;1 ˇı : 0 WD D 20 D b2q3 a ˙ Now we ﬁnd c; d ޚ such that ˛ a bq G.q/, c is a positive odd prime 2 WD cq d 2 not dividing q, and GCD.b 1; c 1/ 2. D i By Lemma 5.6, we know there are ıi i .qޚ /, 1.qޚ/, ' X5.q/, 2 10 2 0 2 X4.q/, and Y5.q/ such that 0 2 0 2 Â Ã 1;2 2 1;2 ˛ . q/ ˛ ı3'ı2 q ' : 0 WD 0 10 D b2q3 a ˙ Conjugating, if necessary, this equality by the matrix diag. 1; 1/, which leaves i i invariant the sets i .qޚ /, i .qޚ /, X5.q/, X4.q/, and Y5.q/, we can assume that the matrices ˛ and ˇ have the same last row. Then ˛ ˇ 1 belongs to 0 0 100 D 0 0 1.qޚ/, and 100ˇ0 ˛0; hence D Â Ã 2 ˇı2 ı ˛ ı ' ı : 100 D 10 0 30 0 20 30 0 300 D b2q a ˙ 1;2 1 1 Now we set ı2 q , ı ı , , etc. WD 100 20 D 200 10 D 0 a b LEMMA 5.8. Let ˛ G.q/ with m 1 an integer. Then there are i D6 2 ıi i .qޚ /, 6 6.qޚ /; X4.q/, ';' X5.q/, and Y5.q/ such that 2 2 2 0 2 2 Â Ã 2m ı1˛ ı5'ı4 6' ı3 : 0 D b a2m ˙ Proof. As in the proof of Lemma 1.2, Â Ã m f ga gb 2 ˇ ˛ f 12 g˛ C and f 1 gޚ: WD D C D 2 i 3 By Lemma 5.5, there are ıi i .qޚ /, 3 3.qޚ /, and ' X5.q/ such 2 2 2 that Â 2 Ã 2 .f ga/ gb ı1ˇ ı5'ı4 3 ˇ C ˙ : DW 0 D 1006 LEONID VASERSTEIN

i Now by Lemma 5.1 with e g, there are ı ; ı i .qޚ / and ' X5.q/ D i i0 2 0 2 such that Â Ã ˇ ı ' ı2 ˇ : 0 30 0 DW 00 D b .f ga/2 ˙ C Since .f ga/2 a2m mod b, we have C Á Â Ã ˇ ı with ı 1.qbf Z/. 00 10 D b a2m 10 2 ˙ Now we set 6 3ı and ı3 ı2ı to ﬁnish our proof. WD 30 WD 10 PROPOSITION 5.9. 1;2 G.q/ C6X5D4Y5C6X5C6X4C5Y5C4X5D6Y5D6X4C3X5D2X5q Y5C2; D i i where Di i .qޚ /, Ci i .qޚ /, X5 X5.q/, Y5 Y5.q/, X4 X4.q/. D D D D D Proof. Let ˇ G.q/. By Corollary 5.7, 2 2 1;2 ˛ D2ˇD2Y5. q/ X5C2X5C3 2 1 1 or (using that D2i C2i and D2i 1 D2i 1/ D D Â Ã 2 1;2 a bq ˇ C2˛ C3X5D2X5q Y5C2; with ˛ , 2 D cq d primes b ; c not dividing q, and GCD. b 1; c 1/ 2. We pick positive j j j j j j j j D m . b 1/ޚ and n . c 1/ޚ such that n m 1. Then a2m 1 mod bq, 2 j j 2 j j D Á and a2n 1 mod cq and n m 1. Á D By Lemma 5.8, Â Ã 2m 1 D1X4˛ D5X5D4Y5C6X5D3: D b a2m 2 ˙ 2m Since a 1 mod b, we obtain easily that 1 D3. So Á 2 2m ˛ X4D1D3D3X5D6Y5C4X5D5 X4D6X5D6Y5C4X5D5: 2 D 2m Hence ˛ D5X5D4Y5C6X5C6X4: 2 T 2n Similarly .˛ / X4D6X5D6Y5C4X5D5: 2D 2n Hence ˛ C5Y5C4X5D6Y5D6X4: 2D Therefore 2m 2n 1;2 ˇ C2.˛ ˛ /C3X5D2X5q Y5C2 2 1;2 C2.D5X5D4Y5C6X5C6X4/.C5Y5C4X5D6Y5D6X4/C3X5D2X5q Y5C2 1;2 C6X5D4Y5C6X5C6X4C5Y5C4X5D6Y5D6X4C3X5D2X5q Y5C2: D We used that C2D5 C6. D POLYNOMIAL PARAMETRIZATION 1007

Counting parameters yields the following result:

COROLLARY 5.10. G.q/ is a polynomial family with 93 parameters. Also, there are polynomials fi ޚŒy1; : : : ; y93 such that 2 Â 2 Ã 1 q f1 qf2 93 ˛ C 2 SL2.ޚŒy1; : : : ; y93/ and ˛.ޚ / G.q/: WD qf3 1 q f4 2 D C Now to prove Theorem 13. Consider an arbitrary principal congruence sub- 2 group SL2.qޚ/. The factor group SL2.qޚ/= SL2.q ޚ/ is commutative, so it is 2;1 2;1 easy to see that it is generated by the images of G.q/ and 1 1.qޚ/. 1/ . Using Corollary 5.11, we conclude that SL2.qޚ/ is a polynomial family with 94 parameters. More precisely:

COROLLARY 5.11. SL2.qޚ/ is a polynomial family which has 94 parameters. Moreover, there are polynomial fi ޚŒy1; : : : ; y94 such that 2 Â Ã 1 qf1 qf2 94 ˛ C SL2.ޚŒy1; : : : ; y94/ and ˛.ޚ / SL2.qޚ/: WD qf3 1 qf4 2 D C Example 5.12. Let H be the subgroup of SL2 ޚ in Example 14. The group G.2/ is a normal subgroup of index 4 in H. The group H is generated by G.2/ 2;1 2;1 together with the subgroup . 1/ 1.ޚ/1 . So H is a polynomial family with 94 parameters.

PROPOSITION 5.13. Every polynomial family H ޚk has the “strong ap- . / . / proximation” property that if t ޚ with t 2, ps 1 ; : : : ; ps t are powers of 2 1 t distinct primes pi , and hi H for i 1; : : : ; t, then there is an h H such that s.i / 2 D 2 h hi mod p for i 1; : : : ; t. Á i D N Proof. Suppose H ˛.ޚ / with ˛ ޚŒy1; : : : ; yN . D s.1/ 2s.t / Let t ޚ with t 2, let p ; : : : ; p be powers of distinct primes pi , and 2 1 t let hi H for i 1; : : : ; t. 2 D .i / .i / N We have hi ˛.u / for i 1; : : : ; t with u ޚ . By the Chinese Re- D DN .i /2 s.i / mainder Theorem, there is a u ޚ such that u u mod pi for i 1; : : : ; t. 2 s.i / Á D Set h ˛.u/. Then h hi mod p for i 1; : : : ; t. D Á i D COROLLARY 5.14. Let H be a subgroup of SL2 ޚ generated by SL2.6ޚ/ and 0 1 the matrix 1 0 . Then H is not a polynomial family. Proof. We do not have the strong approximation property for H. Namely, take t 2, p1 2, p2 3, and s.1/ s.2/ 1. The image of H in SL2.ޚ=2ޚ/ D D D D D is a cyclic group of order 2, and the image of H in SL2.ޚ=3ޚ/ is a cyclic group of order 4. The strong approximation for H (see Proposition 5.13) would imply that the order of the image of H in SL2.ޚ=6ޚ/ is at least 8, while the image is in fact a cyclic group of order 4. 1008 LEONID VASERSTEIN

COROLLARY 5.15. Let X ޚ be an inﬁnite set of positive primes. Then X is not a polynomial family.

Proof. Suppose X is a polynomial family. Let p1 and p2 be distinct primes in X. By Proposition 5.13, there is a z X such that z p1 mod p1 and z 2 Á Á p2 mod p2. Then z is divisible by both p1 and p2 and hence is not prime. This contradiction shows that X is not a polynomial family. Remark 5. By[12, Part VIII, Prob. 97], no nonconstant integer-valued polynomial with rational coefﬁcients takes only prime values. This fact implies easily Corollary 5.15. Two easy solutions to the problem are given, and the result is attributed by Euler to Goldbach.

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(Received January 16, 2006) (Revised October 13, 2006)

E-mail address: [email protected] DEPARTMENT OF MATHEMATICS,PENN STATE UNIVERSITY,UNIVERSITY PARK, PA 16802-6401, UNITED STATES