$B$-orderings for all ideals $B$ of Dedekind domains and generalized factorials
Jeffrey C. Lagarias, Wijit Yangjit
TL;DR
The paper extends Bhargava's theory of $\,\mathfrak{p}$-orderings to all ideals $\frak{b}$ in Dedekind domains by introducing $\frak{b}$-orderings, $\frak{b}$-exponent sequences, and generalized factorials and binomial ideals indexed by arbitrary ideal collections. The core technical advance is a reduction to Bhargava's local $t$-invariant framework via congruence-preserving maps to $K[[t]]$ (Property C) and a Dedekind-domain embedding that ensures well-definedness for all $\frak{b}$, including $(0)$ and $(1)$. The results yield generalized factorials $[k]!_{S,\mathcal{T}}^{h,\{r\}}$ and binomial ideals ${k+\ell \brack \ell}_{S,\mathcal{T}}^{h,\{r\}}$ with integrality and divisibility properties, and they provide a structural framework for studying integer-valued polynomials and related rings over Dedekind domains. These invariants offer a new degree of freedom in Bhargava's theory and open avenues for exploring combinatorial and algebraic interpretations, digit-expansion methods, and extensions to wider ring classes.
Abstract
This paper extends Bhargava's theory of $\mathfrak{p}$-orderings of subsets $S$ of a Dedekind ring $R$ valid for prime ideals $\mathfrak{p}$ in $R$. Bhargava's theory defines for integers $k\ge1$ invariants of $S$, the generalized factorials $[k]!_S$, which are ideals of $R$. This paper defines $\mathfrak{b}$-orderings of subsets $S$ of a Dedekind domain $D$ for all nontrivial proper ideals $\mathfrak{b}$ of $D$. It defines generalized integers $[k]_{S,T}$, as ideals of $D$, which depend on $S$ and on a subset $T$ of the proper ideals $\mathscr{I}_D$ of $D$. It defines generalized factorials $[k]!_{S,T}$ and generalized binomial coefficients, as ideals of $D$. The extension to all ideals applies to Bhargava's enhanced notions of $r$-removed $\mathfrak{p}$-orderings, and $\mathfrak{p}$-orderings of order $h$.