Gelfond-Schneider Theorem/Lemma 1

Lemma
Let $a_1 \left({t}\right), \ldots, a_n \left({t}\right)$ be non-zero polynomials in $\R \left[{t}\right]$ of degrees $d_1, \ldots, d_n$ respectively.

Let $w_1, \ldots, w_n$ be pairwise distinct real numbers.

Then:
 * $\displaystyle F \left({t}\right) = \sum_{j \mathop = 1}^n a_j \left({t}\right) e^{w_j t}$

has at most $d_1 + \cdots + d_n + n − 1$ real roots (counting multiplicities).

Proof
By multiplying through by $e^{−w_n t}$ if necessary, we may suppose that $w_n = 0$ and that otherwise $w_j \ne 0$.

Let $k = d_1 + \cdots + d_n + n$.

We use strong induction on $k$.

If $k = 1$, then $n = 1$ and $d_1 = 0$, and the lemma easily follows.

Let $l \ge 2$ be such that the lemma holds whenever $k < l$.

Suppose $k = l$.

Let $N$ be the number of real roots of $F \left({t}\right)$.

By Rolle's Theorem, the number of real roots of $F' \left({t}\right)$ is at least $N − 1$.

On the other hand:
 * $\displaystyle F' \left({t}\right) = \sum_{j=1}^n b_j \left({t}\right) e^{w_j t}$

where:
 * $b_j \left({t}\right) = a'_j \left({t}\right) + w_j a_j \left({t}\right)$

Note that for $1 \le j \le n − 1$, we have that $b_j \left({t}\right)$ has degree exactly $d_j$.

Also, since $w_n = 0$, either:
 * there are only $n−1$ non-zero polynomials $b_j \left({t}\right)$ in the expression for $F' \left({t}\right)$ above

or:
 * there are $n$ such polynomials and the degree of $b_n \left({t}\right)$ is one less than the degree of $a_n \left({t}\right)$.

We get from the induction hypothesis that $F' \left({t}\right)$ has at most $d_1 + \cdots + d_n + n − 2$ real roots.

Hence, $N − 1 \le d_1 + \cdots + d_n + n − 2$, and the result follows.