Square-free modulus conjecture for polynomial zero counts

Complete proof of the square-free distribution conjecture, with sharp stabilization and an additional prime-power theorem.

Priority: The expected-value clause was already proved, in the greater generality of finite commutative rings, by Ritik Jain, arXiv:2409.06866v2, Theorem 2. The results proved here beyond that earlier theorem are the exact square-free probability distribution, the sharp stabilization threshold for every modulus, and the outstanding prime-power variance.

Let

$$R=\mathbb Z/m\mathbb Z,\qquad f(X_1,\ldots,X_n) = \sum_{0\le a_1,\ldots,a_n\le d} C_{a_1,\ldots,a_n}X_1^{a_1}\cdots X_n^{a_n},$$

where all coefficients are independent and uniform in $R$, and write

$$K_{n,d}^{m}=\#\{x\in R^n:f(x)=0\}.$$

If $m=p_1\cdots p_s$ is square-free, the Chinese remainder theorem identifies the coefficient vector with independent uniform coefficient vectors modulo the $p_i$. It also identifies the zero set with the Cartesian product of the corresponding field zero sets. Hence, for every $d$,

$$K_{n,d}^{m} \stackrel{\mathrm{law}}= \prod_{i=1}^{s}K_{n,d}^{p_i},$$

with independent factors.

If

$$d\ge\max_{1\le i\le s}(p_i-1),$$

evaluation over $\mathbb F_{p_i}$ is surjective onto all functions

$$\mathbb F_{p_i}^{\,n}\longrightarrow\mathbb F_{p_i}.$$

A uniform coefficient vector therefore gives independent uniform values at the $p_i^n$ input points. Consequently

$$\boxed{ K_{n,d}^{m} \stackrel{\mathrm{law}}= \prod_{i=1}^{s}B_i, \qquad B_i\sim\operatorname{Bin}(p_i^n,p_i^{-1}) \text{ independently}. }$$

In particular, its exact probability generating function is

$$\mathbb E[t^{K_{n,d}^{m}}] = \sum_{j_1=0}^{p_1^n}\cdots\sum_{j_s=0}^{p_s^n} t^{j_1\cdots j_s} \prod_{i=1}^s \binom{p_i^n}{j_i} p_i^{-j_i}(1-p_i^{-1})^{p_i^n-j_i}. \tag{1}$$

This proves the conjectured degree stability and answers the source's accompanying probability-generating-function question.

More generally, for any integer $m\ge2$, define

$$\mu(m)=\min\{r\ge1:m\mid r!\}.$$

The monic polynomial

$$Q_r(X)=X(X-1)\cdots(X-r+1),\qquad r=\mu(m),$$

vanishes on every residue modulo $m$, since

$$Q_r(x)=r!\binom{x}{r}.$$

Division by $Q_r$ separately in every variable shows that every polynomial function on $R^n$ has a representative of degree at most $r-1$ in each variable. Therefore the entire distribution of $K_{n,d}^{m}$ stabilizes for

$$\boxed{d\ge\mu(m)-1.}$$

This threshold is sharp. If $d<\mu(m)-1$, the function $X_1^{d+1}$ cannot have a representative of degree at most $d$: otherwise subtraction and restriction to the first coordinate would give a monic null polynomial of degree $d+1$, whose $(d+1)$-st forward difference forces $m\mid(d+1)!$, a contradiction. Thus the evaluation images $I_d\subsetneq I_{d+1}$ are strictly increasing, and

$$\Pr(K_{n,d}^{m}=m^n)=|I_d|^{-1} > |I_{d+1}|^{-1} = \Pr(K_{n,d+1}^{m}=m^n).$$

For square-free $m$, $\mu(m)=\max_{p\mid m}p$, recovering the exact conjectured threshold.

Finally, for every $d\ge1$, put

$$g(h)=\gcd(m,h_1,\ldots,h_n),\qquad h\in R^n.$$

The linear coefficients make $f(x+h)-f(x)$ uniform on the ideal $g(h)R$, while the independent constant coefficient makes $f(x)$ uniform. Hence

$$\Pr(f(x)=f(x+h)=0)=\frac{g(h)}{m^2},$$

and therefore

$$\operatorname{Var}(K_{n,d}^{m}) = m^{n-2}\sum_{h\in R^n}g(h)-m^{2n-2}.$$

For $n=1$ and $m=p^a$, this simplifies to

$$\boxed{ \operatorname{Var}(K_{1,d}^{p^a}) = a\left(1-\frac1p\right) \qquad(d\ge1). }$$

Thus the separate prime-power Conjecture 5.7 is also proved, including its variance and degree stabilization; the mean remains credited to Jain's earlier theorem.

The original conjectures are in Jain–Moon–Wu, Involve 18 (2025), 707–718, Conjectures 5.5 and 5.7.