Abstract
Full Text
UDC 511
MATHEMATICS
V. G. SPRINDZHUK
FINITENESS OF THE NUMBER OF RATIONAL AND ALGEBRAIC POINTS ON SOME TRANSCENDENTAL CURVES
(Presented by Academician I. M. Vinogradov, 13 II 1967)
Studying the Diophantine properties of the values of entire functions, C. Siegel \((^1)\) introduced a class of functions \(\varphi(z)\), which he called \(E\)-functions:
\[ \varphi(z)=\sum_{\nu=0}^{\infty} c_\nu \frac{z^\nu}{\nu!}; \tag{1} \]
(a) the numbers \(c_\nu\) are algebraic and belong to a field \(K\) of finite degree over the field of rational numbers \(Q\); (b) for every \(\varepsilon>0\), \(\overline{|c_\nu|}<c(\varepsilon)(\nu!)^\varepsilon\), where \(\overline{|x|}\) for \(x\in K\) denotes the maximum of the moduli of all quantities conjugate to \(x\) in \(K\); (c) if \(q_m\) is the least natural number for which \(c_\nu q_m\) \((\nu=0,1,\ldots,m)\) are integers, then \(q_m<c(\varepsilon)(m!)^\varepsilon\) for every \(\varepsilon>0\).
Siegel established \((^2)\) the transcendence and algebraic independence of values at algebraic points of \(E\)-functions satisfying a system of linear homogeneous differential equations with polynomial coefficients, under the additional, difficult-to-verify condition of “normality” of the system of these equations. Siegel’s method was substantially developed by A. B. Shidlovskii \((^{4-6})\), who studied the values of \(E\)-functions satisfying a system of nonhomogeneous linear differential equations with polynomial coefficients, replacing Siegel’s “normality” condition by the obvious necessary condition of algebraic independence of the system of \(E\)-functions over the field of rational functions, thereby obtaining a final result. The Siegel–Shidlovskii theory describes the arithmetic nature of values at algebraic points of a great many entire functions (\(\sin z\), \(\cos z\), \(e^z\), the Bessel function \(J_0(z)\), the confluent hypergeometric function, the incomplete gamma function, Fresnel integrals, etc.). Meanwhile, it is known that any weakening of the conditions defining \(E\)-functions completely deprives one of the possibility of carrying out arguments by Siegel’s method. For example, the function
\[ \varphi_\lambda(z)=\sum_{n=0}^{\infty}\frac{z^n}{(\lambda+1)\ldots(\lambda+n)},\qquad \lambda\ne -1,-2,\ldots, \tag{2} \]
which satisfies the differential equation \(zy'+(\lambda-z)y-\lambda=0\) and, for rational \(\lambda\), belongs to the class of \(E\)-functions, for quadratic irrational \(\lambda\) does not belong to this class, and the arithmetic nature of its values for such \(\lambda\) is not decided by Siegel’s method. The fact that \(\varphi_\lambda(z)\), for rational \(\lambda\ne -1,-2,\ldots\), belongs to the class of \(E\)-functions follows from a general theorem of Siegel \((^2)\): the functions
\[ \varphi(z)=\sum_{n=0}^{\infty} \frac{[\alpha_1,n]\ldots[\alpha_l,n]}{[\beta_1,n]\ldots[\beta_m,n]}z^{tn}, \qquad t=m-l\ge 1, \tag{3} \]
\[ [\alpha,0]=1,\quad [\alpha,n]=\alpha(\alpha+1)\ldots(\alpha+n-1)\quad (n=1,2,\ldots) \]
with rational \(\alpha_j,\beta_j\ne0,-1,-2,\ldots\) belong to the class of \(E\)-functions and satis-
satisfy linear differential equations with polynomial coefficients. Siegel called these functions “hypergeometric.”
Shidlovskii \((^{6,7})\) has repeatedly drawn attention to the importance of investigating the Diophantine properties of entire functions that do not belong to the class of Siegel \(E\)-functions. In particular, quite recently Shidlovskii himself \((^8)\) established the irrationality of all, with the exception of a finite number, of the values of the functions \(\varphi_\lambda(z)\), \(K_\lambda(z)\), \(K'_\lambda(z)\), where
\[ K_\lambda(z)=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!(\lambda+1)\cdots(\lambda+n)} \left(\frac{z}{2}\right)^{2n}, \qquad \lambda\ne -1,-2,\ldots, \]
for algebraic \(\lambda\), \(z\in \mathbf K\), \(\mathbf K\) being a fixed field of algebraic numbers.
We introduce the class of \(E^*\)-functions as the class of functions of the form (1), weakening Siegel’s conditions (b) and (c) (but assuming that (a) is satisfied): \((b^*)\) there exists a number \(a\), \(0\le a<1\), for which \(\overline{|c_\nu|}<c(\varepsilon)(\nu!)^{a+\varepsilon}\), where \(\varepsilon>0\) is arbitrary; \((c^*)\) for fixed \(n\), let \(q_{nm}\) be the smallest natural number \(q\) for which all the numbers \(qc_{\nu_1}c_{\nu_2}\cdots c_{\nu_n}\), \(\nu_1+\nu_2+\cdots+\nu_n\le m\), are integers; there exists a sequence of numbers \(\beta_n\), \(0\le \beta_n=o(\sqrt n)\) as \(n\to\infty\), for which \(q_{nm}<c(n,\varepsilon)(m!)^{\beta_n+\varepsilon}\) for any \(\varepsilon>0\).
It is clear that when \(a=\beta_n=0\) \((n=1,2,\ldots)\) the function \(\varphi(z)\) will be an \(E\)-function, so that the class of \(E^*\)-functions contains Siegel’s class of \(E\)-functions. It turns out that, applying the original Thue–Siegel idea for the construction of approximating forms, and then making a direct passage to numerical values without the construction and investigation of a system of linearly independent forms required in Siegel’s method, one can obtain some results on the Diophantine properties of values of \(E^*\)-functions satisfying linear differential equations with polynomial coefficients. By this method, at the present stage of its development, it is not possible to establish the transcendence of values of \(E^*\)-functions, although assertions of the following type are proved:
(A) For any natural number \(d\) and any field of algebraic numbers \(\mathbf K_1\) of finite degree over \(Q\), there exists only a finite number of \(z\in \mathbf K_1\) for which the function \(\varphi(z)\) assumes algebraic values of degree at most \(d\) over \(Q\). In particular, on the curve \(y=\varphi(z)\) there lie no more than finitely many points \((y,z)\in \mathbf K_1\times \mathbf K_1\).
Theorem 1. If a transcendental \(E^*\)-function \(\varphi(z)\) satisfies the differential equation
\[ p(z)y'+q(z)y+r(z)=0, \tag{4} \]
where \(p(z)\), \(q(z)\), \(r(z)\) are polynomials with algebraic coefficients, then assertion (A) holds for \(\varphi(z)\).
Theorem 2. Every transcendental function of the form
\[ \varphi(z)=e^{Q(z)}\left(\int_0^z R(x)e^{-Q(x)}\,dx+c\right), \]
where \(Q(z)\), \(R(z)\) are polynomials with algebraic coefficients and \(c\) is an arbitrary complex number, satisfies condition (A).
Theorem 3. Siegel hypergeometric functions (3) with algebraic parameters \(\alpha_i,\beta_j\ne 0,-1,-2,\ldots\) belong to the class of \(E^*\)-functions. In particular, this is true for the functions (2) with algebraic \(\lambda\), so that (A) holds for them.
Theorem 1 is proved according to the following scheme. Applying Lemma II from \((^3)\), p. 67, one can, for any \(h>h(n,\varepsilon)\), construct a system of polynomials \(P_j(z)\) with integer coefficients from \(\mathbf K\), of degree at most \(h\), satisfying the conditions:
\[ 0\ne H=\max_{0\le j\le n}\overline{|P_j(z)|}<H_0^{2(\alpha+\beta n)+\varepsilon},\qquad H_0=h^h, \tag{5} \]
function
\[ f(z)=\sum_{j=0}^n P_j(z)\varphi^j(z) \tag{6} \]
at \(z=0\) has a zero of order \(m\ge h[\sqrt n]\). Suppose that the function \(f(z)\) has, at the points \(z=x_i\ne0\), zeros of orders \(m_i\), respectively \((i=1,2,\ldots,k)\), \(m_1=\max_{(i)}m_i\). Applying the integral representation
\[ f^{(m_1)}(x_1)=\frac{m_1!}{(2\pi i)^2} \int_{|z-x_1|=1}\frac{dz}{(z-x_1)^{m_1+1}}\times \]
\[ \times\int_{|\xi|=R} \left(\frac{z}{\xi}\right)^m \left(\frac{z-x_1}{\xi-x_1}\right)^{m_1} \cdots \left(\frac{z-x_k}{\xi-x_k}\right)^{m_k} \frac{f(\xi)}{\xi-z}\,d\xi \]
and choosing a suitable \(R>\max_{(i)}|x_i|=\rho\), we find
\[ 0\ne |f^{(m_1)}(x_1)|<H_0^{\,1-([\sqrt n]-1)/v+\varepsilon}\,m_1^{-(k/v-1)m_1+\varepsilon}, \tag{7} \]
where \(v=1/(1-\alpha)\) is the order of the function \(\varphi(z)\), \(h>h(n,\varepsilon,\rho)\). The polynomials \(p,q,r\) in equation (4) may be regarded as polynomials with integral coefficients over \(K\), of degrees, say, \(\le s\), \(\overline{|p|}+\overline{|q|}+\overline{|r|}\le a\). Define the polynomials \(P_{jl}=P_{jl}(z)\) by the equalities \(P_{j0}=P_j\) \((j=0,1,\ldots,n)\), where the \(P_j\) are determined by conditions (5), (6), and \(P_{jl}\) \((l=1,2,\ldots)\) by the equalities
\[ P_{0l+1}=pP'_{0l}-rP_{1l}, \]
\[ P_{jl+1}=pP'_{jl}-jqP_{jl}-(j+1)rP_{j+1,l} \qquad (j=1,2,\ldots,n-1), \]
\[ P_{nl+1}=pP'_{nl}-nqP_{nl}. \]
Then
\[ \max_{0\le j\le n}\overline{|P_{jl}|}<a^lH(h+sl+2n+1)^l. \tag{8} \]
Putting \(f_0(z)=f(z)\), \(f_{l+1}(z)=p(z)f_l'(z)\), we find
\[ f_l(z)=\sum_{j=0}^n P_{jl}(z)\varphi^j(z)\quad (l=0,1,\ldots),\qquad f_{m_1}(x_1)=p^{m_1}(x_1)f^{(m_1)}(x_1). \]
If we assume that the numbers \(x_1,x_2,\ldots,x_k\) are algebraic and belong to the field \(K_1\), then, in view of (5), (7), (8), we can establish (A), if \(n\) is taken sufficiently large and it is assumed that \(k\) can be sufficiently large.
Passing from equation (4) to the system of equations
\[ y_l'=\sum_{j=1}^m Q_{jl}y_j+Q_l\qquad (l=1,2,\ldots,m), \tag{9} \]
where \(Q_{jl}, Q_l\) are rational functions with algebraic coefficients, one can prove, for a system of \(E^*\)-functions, assertions of the following type:
(B) For any natural number \(d\) and any field of algebraic numbers \(K_1\) of finite degree over \(Q\), there exists only a finite number of \(z\in K_1\) for which the functions \(\varphi_1(z),\ldots,\varphi_m(z)\) simultaneously take algebraic values of degrees not exceeding \(d\) over \(Q\). In particular, on the curve \(\Gamma=(\varphi_1(z),\ldots,\varphi_m(z))\) there lie no more than a finite number of points from \(K_1^m\) for \(z\in K_1\).
Theorem 4. If the \(E^*\)-functions \(\varphi_1(z),\ldots,\varphi_m(z)\) satisfy the system of equations (9) and are algebraically independent over the field of rational
functions \(C(z)\), \(C\) being the field of complex numbers, then they satisfy condition (B).
Theorem 5. If all the coefficients of equation (9) are polynomials, then for any system of functions \(\varphi_1(z), \ldots, \varphi_m(z)\) satisfying (9) and algebraically independent over \(C(z)\), assertion (B) is valid.
The proof of Theorem 4 is carried out according to the same scheme as the proof of Theorem 1, and contains only technical complications. The proof of Theorem 2 and of the more general Theorem 5 is based on the following initial remarks. If the functions \(\varphi_1(z), \ldots, \varphi_m(z)\) at an algebraic point \(\varkappa\) take algebraic values simultaneously, then, by virtue of (9), we can establish that the functions \(\varphi_1(z+\varkappa), \ldots, \varphi_m(z+\varkappa)\) belong to the class \(E^*\). For the proof of \((6^*)\) it should be noted that equation (9) is satisfied only by entire functions of finite order (by the condition of Theorem 5, \(Q_{jl}, Q_l\) are polynomials), and the proof of \((B^*)\) is obtained by successive differentiation of (9), with simple estimates. Then we apply Theorem 4.
Finally, both the membership of Siegel hypergeometric functions with algebraic parameters in the class of \(E^*\)-functions, and the nonmembership of the function (2) with quadratic irrationality in the class of \(E^*\)-functions, are proved by the application of \(p\)-adic analysis.
Institute of Mathematics
Academy of Sciences of the BSSR
Received
8 II 1967
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