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Source. Bajpai--Bennett--Chan, accepted author manuscript (June 26, 2023), Lemma 3.1, pp. 7--8. The manuscript presents this lemma as essentially Lemma 8.1 of Shorey and Tijdeman, Arithmetic properties of blocks of consecutive integers, in From Arithmetic to Zeta-functions (Springer, 2016), 455--471 (DOI), and proves it for completeness (p. 7).

Statement. For integers $\ell\geq2$ and $d\geq1$, put

$$F_d(X)= \prod_{\substack{1\leq j\leq\ell\\j\ {\rm odd}}} (X+jd)^{\binom\ell j} - \prod_{\substack{0\leq j\leq\ell\\j\ {\rm even}}} (X+jd)^{\binom\ell j}.$$

Then $F_d(X)=d^\ell G_d(X)$, where $G_d(X)$ is an integral homogeneous binary form of degree $2^{\ell-1}-\ell$, and $G_0(1)=(\ell-1)!$.

Proof. Both products have total degree

$$\sum_{j\ {\rm odd}}\binom\ell j =\sum_{j\ {\rm even}}\binom\ell j=2^{\ell-1},$$

so $F_d(X)$ is homogeneous in $X,d$ of that degree. Consider

$$R(x)= \frac{\prod_{j\ {\rm odd}}(1+jx)^{\binom\ell j}} {\prod_{j\ {\rm even}}(1+jx)^{\binom\ell j}}.$$

Expanding its logarithm gives

$$\log R(x)= \sum_{i\geq1}\frac{(-1)^{i-1}x^i}{i} \sum_{j=1}^{\ell}(-1)^{j-1}\binom\ell j j^i. \tag{1}$$

By inclusion-exclusion, the inner sum is zero for $i<\ell$: it is, up to sign, the number of surjections from an $i$-element set onto an $\ell$-element set. When $i=\ell$, it is $(-1)^{\ell-1}\ell!$. The signs in (1) therefore give

$$\log R(x)=(\ell-1)!x^\ell+O_\ell(x^{\ell+1}),$$

and exponentiation yields the same leading nonconstant term for $R(x)$. Substitute $x=d/X$. The difference of the numerator and denominator is

$$F_d(X)=(\ell-1)!X^{2^{\ell-1}-\ell}d^\ell +O_\ell\!\left(X^{2^{\ell-1}-\ell-1}d^{\ell+1}\right).$$

Thus every homogeneous monomial of $F_d$ contains $d^\ell$, and its coefficient at $X^{2^{\ell-1}-\ell}d^\ell$ is $(\ell-1)!$. Division by $d^\ell$ proves both integrality and the stated degree and leading value of $G$.

Used by. Theorem 1.1.

Bears on. #937.