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Source. Theorem 3, p. 1172, of P. Erdős, "Some remarks on polynomials," Bull. Amer. Math. Soc. 53 (1947), 1169-1176. Pages are the journal's own, as on the source card.

Context

The paper quotes (p. 1172) a result of Schur (Math. Z. 1 (1918), 377-402, pp. 389-391): if $a_0x^n+\cdots+a_n$ has integer coefficients and all its roots are in $(-1,+1)$ and distinct, then $\lvert a_0\rvert>(2^{1/2}-\epsilon)^n$ for sufficiently large $n$. Theorem 3 is presented as a stronger theorem.

Statement

Theorem 3 (p. 1172), as printed: "Let $f_n(x)=a_0x^n+\cdots+a_n$ be a polynomial with integer coefficients and $f_n(-1)\neq0$, $f_n(0)\neq0$, $f_n(+1)\neq0$. Then, $\lvert a_0\rvert\ge2^{n/2}$."

The printed statement names no condition on the roots, but the proof needs one: it uses $\lvert(1-x_i^2)x_i^2\rvert\le\tfrac14$ for every root $x_i$, which holds for real roots in $[-1,1]$, the setting of Schur's result just quoted.

The paper calls the bound best possible, citing the example $2^n(x-1/2)^n$ as printed; that polynomial has leading coefficient $2^n$, not $2^{n/2}$. For even $n$ the polynomial $(2x^2-1)^{n/2}$ meets the hypotheses with leading coefficient exactly $2^{n/2}$.

The paper adds (pp. 1172-1173) that using Schur's fact that the discriminant is an integer one obtains $\lvert a_0\rvert>(2^{1/2}+c)^n$ for large $n$, and it quotes a polynomial of degree $2n$ constructed by Schur.

Read depth. Claims checked: the statement, Schur's result as quoted and the two-line proof were read on the print.

Proof pointer

Page 1172. With $x_1,\ldots,x_n$ the roots, the integer $\lvert f_n(-1)f_n^2(0)f_n(+1)\rvert$ equals $\lvert a_0^4\prod_i(1-x_i^2)x_i^2\rvert$ and is nonzero, hence at least $1$; bounding each factor $\lvert(1-x_i^2)x_i^2\rvert$ by $\tfrac14$ gives $\lvert a_0\rvert\ge2^{n/2}$.