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Statement

Theorem 3 (p. 69). Let $L_1,L_2$ be integers with $L_1\ge1$, $L_2\ge1$. Then for every polynomial $A(z,x_1,x_2,x_3)\in\mathbb C[z,x_1,x_2,x_3]$ with $A\ne0$, $\deg_zA\le L_1$ and $\deg_{x_i}A\le L_2$,

$$\operatorname{ord}A\bigl(z,P(z),Q(z),R(z)\bigr)\le cL_1L_2^3,\qquad c=2\cdot10^{45}.$$

Here $\operatorname{ord}\varphi(z)$ is the multiplicity of the zero of $\varphi$ at $z=0$ (p. 69), and $P,Q,R$ are Ramanujan's functions of Theorem 1, which are algebraically independent over $\mathbb C(z)$ by Mahler's 1969 theorem (p. 66). The paper describes Theorem 3 as a measure of that algebraic independence and as playing an important part in the proof of Theorem 1.

Source. Yu. V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65--96 (Russian; English translation Sb. Math. 187 (1996), no. 9, 1319--1348), Theorem 3 (Теорема 3), p. 69 of the Russian original; see the source card. Labels and pages follow the Russian pagination.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof was not read.

Proof pointer

The title of Section 2 (p. 69) announces the reduction of Theorems 1 and 2 to Theorem 3; the proof of Theorem 3 itself occupies the later sections of the paper and was not read here.

Bears on

No Erdős problem directly; it is the zero estimate behind Theorem 1, which bears on #250 through Corollary 2.