Multiplicity $=$ Volume formula and Newton non-degenerate ideals in regular local rings

TL;DR

The paper extends Newton non-degenerate ideals and Newton polyhedra to regular local rings, proving that the integral closure characterizes NND-ness. It introduces the limiting body $\mathcal C(\mathcal I)$ for graded families and proves a criterion equating multiplicity with the $d$-dimensional co-volume of $\mathcal C(\mathcal I)$ whenever the family contains an NND subfamily. This yields a practical alternative to Newton-Okounkov bodies for the Multiplicity=Volume formula and yields structural results for products, intersections, and powers of NND ideals, linking analytic and algebraic perspectives in regular local rings.

Abstract

We develop the notions of Newton non-degenerate (NND) ideals and Newton polyhedra for regular local rings. These concepts were first defined in the context of complex analysis. We show that the characterization of NND ideals via their integral closures known in the analytical setting extends to regular local rings. We use the limiting body $\mathcal{C}(\mathcal{I})$ associated to a graded family $\mathcal{I}$ of ideals to provide a new understanding of the celebrated "Multiplicity $=$ Volume" formula. Particularly, we prove that, for a Noetherian graded family $\mathcal{I}$ of $\mathfrak{m}$-primary ideals in a regular local ring $(R,\mathfrak{m})$ of dimension $d$, the equality $$e(\mathcal{I}) = d!\text{co-vol}_d(\mathcal{C}(\mathcal{I}))$$ holds if and only if $\mathcal{I}$ contains certain subfamily of NND ideals.

Figures (4)

• Figure 1: $\mathop{\mathrm{NP}}\nolimits(I)$ for $I = ((x^2+1)(y^2+2), (y^2+2)^2) \subseteq {\mathbb R}[x,y]$.
• Figure 2: $\Gamma_{\bf x}(I)$ for $I = ((x^2+1)^2(y^2+2)+(x^2+1)(y^2+2)^3) \subseteq {\mathbb R}[x,y]$.
• Figure 3: $\Gamma_{\bf x}(I)$ for $I = (x^4+x^2y^5z^4,y^3z-x^3yz^2+xyz^3) \subseteq R$.
• Figure 4: $\Gamma_{\bf x}(I)$ for $I = (x^4+y^4, xy^2+x^2y) \subseteq {\mathbb C}[[x,y]]$.

Theorems & Definitions (69)

• Definition 2.1: HS08
• Definition 2.2: See HS08
• Example 2.3
• Theorem 2.4: HS08
• Corollary 2.5
• proof
• Corollary 2.6
• proof
• Definition 2.7
• Definition 2.8