On values of linear and quadratic forms at integral points

More documents

Recommendations

Info

6 Complements to the Oppenheim conjectureWe shall now discuss analogous questions about values of quadratic forms atintegral points, in some cases complementary to the results on Oppenheimconjecture.Let us first consider the nondegeneracy condition in the Oppenheim conjecture,as in Theorem 5.1. Though it has been traditional to assume nondegeneracyin the context of the problem, unlike in some other problems thegeneral case here does not readily reduce to proving the result in the caseof nondegenerate forms (the reader may convince himself of this by perusingthe cases arising in the proof of Theorem 6.1 below).Let Q be a (possibly degenerate) quadratic form. We recall that theradical of Q is the subspace W = {w ∈ IR n | Q(v + tw) = Q(v) for all v ∈IR n and t ∈ IR}. Suppose that W is a rational subspace of IR n (namely, itis defined by linear equations with rational coefficients). Then the image( Z n + W )/W of Z n in R n /W is a lattice and IR n /W can be realised as IR m ,where m is the codimension of W in IR n , with ( Z n + W )/W correspondingto Z m . Hence Q factors to a quadratic form, say Q ′ , on IR m and Q( Z n ) =Q ′ ( Z m ). Further, it can be seen that there exists a positive integer r suchthat Q ′ (P( Z m )) ⊆ Q(P( Z n ) ⊆ 1 r Q′ (P( Z m ). This shows in particular thatif the radical of a quadratic form Q is a rational subspace of codimension 2then the study of its values reduces to the 2-variable case; (see below for adiscussion on the 2-variables case). We now note the following:Theorem 6.1. Let Q be a quadratic form on IR n , n ≥ 3, which is indefinite(that is, there exist v, w ∈ IR n such that Q(v) < 0 < Q(w)) and not amultiple of a form with rational coefficients. Then at least one of the followingconditions holds:i) the radical of Q is a rational subspace of codimension 2; orii) Q(P( Z n )) is dense in IR.Proof: If the rank of Q is at least 3 then the argument as in the proof of Theorem1 in [8] (reducing the number of variables to 3, in which case Q would benondegenerate) shows, together with Theorem 5.1, that condition (ii) holds.Note that since the form is indefinite, the rank is at least 2. Now supposethat Q is of rank 2. Then there exist linear forms L 1 and L 2 on IR n such thatQ(v) = L 1 (v)L 2 (v) for all v ∈ IR n . Since Q is indefinite L 1 and L 2 are linearlyindependent (as linear forms). Suppose first that no linear combination10
of the form λ 1 L 1 + λ 2 L 2 , with λ 1 , λ 2 ∈ IR, is a rational linear form. Let F nbe the space of linear forms on IR n , and consider the contragradient actionof SL(n, Z) on F n as in §4. Hence by Corollary 4.2 for L 1 , L 2 as above theset of pairs {(L 1 (x), L 2 (x)) | x ∈ P( Z n )} is dense in IR 2 . Since Q = L 1 L 2this implies that Q(P( Z n )) is dense in IR.Next suppose that there exist λ 1 , λ 2 ∈ IR such that λ 1 L 1 + λ 2 L 2 = L 0 ,say, is a rational linear form. If the span of L 1 and L 2 contains a rationalform which is not a multiple of L 0 then the radical of Q, which is given by{v ∈ IR n | L 1 (v) = L 2 (v) = 0}, is a rational subspace of codimension 2,so condition (i) is satisfied. We may therefore suppose that any rationalform in the span of L 1 and L 2 is a multiple of L 0 . Let Γ 0 be the subgroup{γ ∈ SL(n, Z) | L 0 ◦ γ = L 0 }. Since Q is not a multiple of a form withintegral coefficients, at least one of L 1 and L 2 is not a multiple of a linearform with integral coefficients; for definiteness assume that this holds forL 1 . Then in view of our assumption as above, any rational linear form inthe span of L 1 and L 0 is a multiple of L 0 . Now let t ∈ L 0 (P( Z n )). Thenby Corollary 4.2 the set of pairs {(L 1 (x), t) | x ∈ P( Z n ), L 0 (x) = t} isdense in {(s, t) | s ∈ IR}. We note that λ 2 ≠ 0, since L 0 is a rational formwhile L 1 is not a multiple of any rational form. Now, for any x ∈ IR n ,Q(x) = L 1 (x)L 2 (x) = L 1 (x)(L 0 (x) − λ 1 L 1 (x))/λ 2 . Hence the closure ofQ(P( Z n )) contains the set {s(t − λ 1 s)/λ 2 | s ∈ IR}, which is the whole ofIR if λ 1 = 0 and contains [−t 2 /4|λ 1 λ 2 | , t 2 /4|λ 1 λ 2 |] if λ 1 ≠ 0 (it extends toinfinity on one side, depending on the sign of λ 1 /λ 2 ). Since L 0 (P( Z n )) isunbounded this means that Q(P( Z n )) is dense in IR, thus completing theproof.We now consider the case with n = 2. It was noted earlier that thereexist binary quadratic forms which are not multiples of forms with integercoefficients but nevertheless their values at integer points do not intersectsome interval of the form (0, δ) for some δ > 0. Now let Q be any indefinitebinary quadratic form. Then Q(v, w) can be expressed as (av + bw)(cv + dw)for all v, w ∈ IR (with respect to the standard basis), where a, b, c, d ∈ IR andad − bc ≠ 0. An elementary argument then shows that for every ɛ > 0 thereexist x, y ∈ Z, not both 0, such that |Q(x, y)| < ɛ, if and only if either a/bor c/d is not badly approximable. The question of taking values arbitrarilyclose to nonzero real numbers is however more intricate (see [2]). It is closelyrelated to the dynamics of the flow induced by the one parameter subgroupof diagonal matrices on the space SL(2, IR)/SL(2, Z). We shall however notgo into the details of this here.11
Page 4 and 5: Let N be the subgroup of SL(n, IR)
Page 6 and 7: The corollary implies in particular
Page 8 and 9: 5 The Oppenheim conjectureWe have s
Page 12 and 13: 7 Simultaneous solution of quadrati
Page 14 and 15: |e t x 3 | = |x 1 x 3 |/ɛ ≤ ɛ a
Page 16 and 17: 9 Comments in conclusionThe study o
Page 18: [17] D. Kleinbock and G.A. Margulis

On values of linear and quadratic forms at integral points

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?