Bhargava factorial

Bhargava factorial (also called Bhargava's factorial function) is a generalization of the classical factorial function developed by the Fields Medal-winning mathematician Manjul Bhargava as part of his doctoral thesis at Harvard University in 1996. It associates a positive integer $k!S$ with every positive integer $k$ and any infinite subset $S$ of the integers $\mathbb{Z}$, such that when $S = \mathbb{Z}$, the value $k!{\mathbb{Z}}$ reduces to the ordinary factorial $k!$.

Motivation

The classical factorial appears in several fundamental number-theoretic results, including:

1. For any positive integers $m$ and $n$, $(m+n)!$ is a multiple of $m!,n!$.
2. For a primitive integer polynomial $f(x)$ of degree $k$, the greatest common divisor of the set of values ${f(a) : a \in \mathbb{Z}}$ divides $k!$ (Pólya's theorem).
3. For any integers $a_0, a_1, \dots, a_n$, the product of their pairwise differences is a multiple of $0!,1!\cdots n!$.
4. The number of polynomial functions from $\mathbb{Z}$ to $\mathbb{Z}/n\mathbb{Z}$ is $\prod_{k=0}^{n-1} \frac{n}{\gcd(n, k!)}$.

Bhargava asked whether the set $\mathbb{Z}$ could be replaced by an arbitrary infinite subset $S \subset \mathbb{Z}$ and a function $k!_S$ defined so that these theorems remain true. He answered this question in the affirmative.

Definition via $p$-orderings

Let $S$ be an infinite subset of $\mathbb{Z}$ and let $p$ be a prime. A $p$-ordering of $S$ is a sequence ${a_0, a_1, a_2, \dots}$ of elements of $S$ constructed inductively as follows:

• $a_0$ is any element of $S$.
• For $k \ge 1$, $a_k$ is chosen to minimize the highest power of $p$ dividing the product $(a_k - a_0)(a_k - a_1)\cdots(a_k - a_{k-1})$.

For each $k$, let $v_k(S, p)$ be the highest power of $p$ dividing this product. The sequence ${v_0(S, p), v_1(S, p), v_2(S, p), \dots}$ (with $v_0(S, p) = 1$) is called the associated $p$-sequence of $S$. A key result is that this sequence is independent of the particular choices made in constructing the $p$-ordering—it is an invariant of $S$.

The Bhargava factorial is then defined as:

$$ k!S = \prod{p\text{ prime}} v_k(S, p) $$

Examples

• Set of integers $S = \mathbb{Z}$: $k!_{\mathbb{Z}} = k!$ (the ordinary factorial).
• Set of even integers $S = 2\mathbb{Z}$: $k!_{2\mathbb{Z}} = 2^k \cdot k!$.
• Set of integers of the form $an + b$: $k!_{a\mathbb{Z}+b} = a^k \cdot k!$.
• Set of powers of 2 $S = {1, 2, 4, 8, \dots}$: $k!_S = (2^k - 1)(2^k - 2)\cdots(2^k - 2^{k-1})$.
• Set of squares $S = {0, 1, 4, 9, \dots}$: $k!_S = (2k)!/2$.
• Set of prime numbers $S = \mathbb{P}$: The first few values are $0!{\mathbb{P}} = 1$, $1!{\mathbb{P}} = 1$, $2!{\mathbb{P}} = 2$, $3!{\mathbb{P}} = 24$, $4!{\mathbb{P}} = 48$, $5!{\mathbb{P}} = 5760$, $6!_{\mathbb{P}} = 11520$ (OEIS A053657).

Properties

• Integrality of binomial coefficients: For any subset $S$, the generalized binomial coefficients $\binom{n}{k}_S = \frac{n!_S}{k!_S (n-k)!_S}$ are integers.
• Fixed divisor theorem: For a primitive polynomial $f$ of degree $k$ over $S$, the fixed divisor $d(S, f)$ divides $k!_S$.
• Monotonicity: If $T \subseteq S$, then $k!_S$ divides $k!_T$.
• Generalization to Dedekind domains: The construction extends to arbitrary subsets of Dedekind domains, where the factorials become ideals.

Applications

Bhargava factorials have been used in the theory of integer-valued polynomials (providing a complete answer to Pólya's question on regular bases), $p$-adic interpolation (generalizing Mahler's theorem to arbitrary compact subsets of local fields), and the study of polynomial functions over finite rings. The concept has also been extended to several variables and to more general algebraic settings.