Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 20 / 4

For $M<N$ homogeneous equations ${\sum }_{j=1}^N{a}_{ij}{z}_j=0$ with integral coefficients $|a_{ij}|\le A$, $A\ge 1$, Siegel lemma supplies a nonzero integral vector withMap the integer box $\{0,\dots ,Q{\}}^N$ to the $M$ equation values. There are $(Q+1{)}^N$ inputs, while each output coordinate lies in $[-NAQ,NAQ]$, so there are at most $(2NAQ+1{)}^M$ outputs. Since $Q+1>(2NA{)}^{M/(N-M)}$ and $2NAQ+1\le 2NA(Q+1)$, the input count is strictly larger. Two inputs have the same image; their nonzero difference solves every equation and has the stated norm bound. This proves the lemma by the pigeonhole proof of integer Siegel lemma.

Rational coefficients are handled by clearing denominators. For coefficients in a number field of degree $d$, expansion in a rational basis gives at most $dM$ rational equations; if $N>dM$ the same argument gives an integer solution with the corresponding effective bound after clearing the basis denominators. The lemma concerns homogeneous equations; arbitrary inhomogeneous integer equations need not be soluble.

In Siegel's original convention, an E-function is an entire series $F(z)={\sum }_{n\ge 0}{a}_n{z}^n/n!$ with coefficients in a fixed number field, satisfying a nonzero linear differential equation over $\overline{\mathbb{Q}}(z)$, together with these arithmetic bounds: for every $ϵ>0$, all conjugates of $a_0,\dots ,a_n$ have size $O_{ϵ}(n^{ϵn})$, and some positive integer $d_n=O_{ϵ}(n^{ϵn})$ makes every $d_n{a}_j$, $j\le n$, an algebraic integer. A frequently used stricter convention replaces both bounds by $C^{n+1}$. The original and strict growth definitions should not simply be declared equivalent; the exponential examples below satisfy the strict one.

For sums, work in the compositum of the coefficient fields. Coefficient conjugate bounds add. The product of the two common denominators clears both lists; using $ϵ/2$ in each original denominator bound gives the required $O_{ϵ}(n^{ϵn})$ product bound. Finally all derivatives of $F+G$ lie in the sum of the two finite-dimensional derivative spaces over that rational function field, so they satisfy a nontrivial linear dependence, giving a differential equation for the sum. Thus sums are again E-functions, the addition closure of Siegel E-functions.

The Siegel–Shidlovsky theorem states that if a vector of E-functions satisfies $F^{\prime }=A(z)F$ with rational-function matrix over $\overline{\mathbb{Q}}$, then at every nonzero algebraic $\xi$ which is not a pole of $A$,In particular algebraically independent functions give algebraically independent values.

Take $F_j(z)=e^{{\gamma }_{j}z}$ for algebraic numbers ${\gamma }_j$ linearly independent over $\mathbb{Q}$. Their coefficients are ${\gamma }_j^n$ with exponential conjugate and denominator bounds, and $F_j^{\prime }={\gamma }_j{F}_j$. A polynomial relation among the functions expands into a relation among exponentials with distinct exponents $\sum m_j{\gamma }_j$ and polynomial coefficients after clearing rational-function denominators; exponential-polynomial independence excludes it. At $\xi =1$ the theorem gives algebraic independence of the $e^{{\gamma }_j}$. For arbitrary distinct algebraic ${\alpha }_i$, choose a rational basis of their span and divide that basis by a common denominator. The $e^{{\alpha }_i}$ are distinct Laurent monomials in the corresponding algebraically independent exponential values, so are linearly independent over $\overline{\mathbb{Q}}$. This recovers the general Lindemann–Weierstrass theorem. Its simplest instance uses just $F(z)=e^z$ at nonzero algebraic $\xi$.