# zbMATH — the first resource for mathematics

On the representation of integers by binary forms. (English) Zbl 0102.03601
The most striking result is the following consequence of Theorem 2. Let $$F (x, y)$$ be a binary form of degree $$n$$ with rational integral coefficients and nonzero discriminant such that $$F(1, 0)\neq 0$$, $$F(0, 1) \neq 0$$. Let a be the height of $$F$$ (= maximum of the absolute value of the coefficients and let $$p_1,\dots, p_t$$ be a fixed set of $$t$$ primes. Then
(i) there are at most $$c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}$$ coprime pairs of integers $$x, y$$ for which $$F (x, y)$$ has no prime factors distinct from $$p_1,\dots, p_t$$;
(ii) there are at most $$(c_3 n)^{t+1}$$ coprime pairs of integers $$x, y$$ such that $$F(x,y)$$ is greater than a certain constant depending an $$a$$ and $$n$$ and for which $$F(x,y)$$ has no prime factors distinct from $$p_1,\dots, p_t$$.
(iii) Let $$p$$ be a prime. Then if $$p$$ is sufficiently large there are at most $$c_4 n^2$$ coprime pairs of integers for which $$\pm F(x,y)$$ is equal to $$p$$ or a power of $$p$$. Here $$c_1,c_2,c_3,c_4$$ are constants which can be given explicitly and are not too large.
Theorem 2 itself gives better but very elaborate estimates.
Statement (i) above gives, in particular, the upper bound $$c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}$$ for the number of solutions of $$F (x, y) = m$$, where $$t$$ is the number of prime factors of $$m$$, and this is much better than the estimate obtained by H. Davenporth and K. F. Roth [Mathematika 2, 160–167 (1955; Zbl 0066.29302)]. The proof is an elaboration of an earlier one of K. Mahler [Math. Ann. 107, 691–730 (1933; Zbl 0006.10502; JFM 59.0220.01)] of a less precise result (which as the authors point out is, however in general much stronger than that of Davenport and Roth), and is related closely to Mahler’s work an the $$p$$-adic Thue–Siegel theorem. Of independent interest are Lemma 1 giving a lower bound for $$f (z)$$, where $$f$$ is a polynomial wich complex coefficients, in terms of the height, discriminant and degree of $$f$$ and of the minimum distance of $$z$$ from a zero of $$f$$ which is stronger than that given by N. I. Fel’dman [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 53–74 (1951; Zbl 0042.04802)] and the $$p$$-adic analogue (Lemma 2) which improves an estimate of F. Kasch and B. Volkmann [Math. Z. 72, 367–378 (1960; Zbl 0091.04801)].
Using these the authors obtain a lower bound for $|F (x, y)|\prod_{\tau=1}^t |F(x,y)|_{p_{\tau}}$ in terms of the minimum distance of $$x/y$$ from the complex and $$p$$-adic zeros of $$F (x, 1) = 0$$. When the only prime factors of $$F (x, y)$$ are $$p_1,\dots, p_t$$ then the expression (*) is unity. Hence the machinery of the $$p$$-adic Thue–Siegel theorem applies. There is an elaborate and ingenious subdivision into cases. There is also a rather elaborate generalization of Theorem 2 (Theorem 3) in which, roughly speaking, although $$F (x, y)$$ is not entirely composed of $$p_1,\dots, p_t$$, the product of the primes dividing $$F(x, y)$$ other than these is of lower order of magnitude.
Reviewer: J. W. S. Cassels

##### MSC:
 1.1e+17 General binary quadratic forms
number theory
Full Text: