Title: On polynomials associated to Voronoi diagrams of point sets and crossing numbers
Domain: Math

Question: How can one rigorously establish sharp upper bounds on the Hilbert–Samuel multiplicity $e(R)$ for Cohen–Macaulay local rings of prime characteristic, thereby extending known results from the Gorenstein case to general CM rings? In particular, under what conditions can these bounds be expressed in terms of the dimension, embedding dimension, type, and Frobenius test exponent, and how do the proofs unfold in both odd and even dimensions? Furthermore, how do special classes such as F-pure, F-rational, and F-nilpotent rings refine these results, what technical tools from tight closure theory and the Briançon–Skoda framework are required, and how does the Frobenius action on local cohomology modules provide the mechanism controlling multiplicity bounds?* | |

Background: 1. Basic Setup: | - (R,m): Noetherian local ring of characteristic p | - Dimension: d | - Embedding dimension: v | - Type: s (for Cohen-Macaulay rings) | | 2. Historical Development: | - Huneke–Watanabe (2015): proved that e(R) ≤ (v/d) for F-pure rings. | - Katzman–Zhang (2019): extended the result, showing e(R) ≤ Q^(v-d)(v/d). | - More recent work: extends these bounds further to general Cohen–Macaulay (CM) rings. | | 3. Frobenius Operations and Test Exponents: | - Frobenius Closure: I^F = { x | x^(p^e) ∈ I^[p^e] for some e }. | - For a parameter ideal q, its Frobenius closure is denoted q^F. | - Test Exponent Framework: | - Fte(R): the least e such that (q^F)^[p^e] = q^[p^e]. | - HSL(R): Hartshorne–Speiser–Lyubeznik number. | - Relationship: for CM rings, Fte(R) = HSL(R). | | 4. F-Singularity Classes and Properties: | - Key Classes: | - F-pure: Frobenius map is pure. | - F-rational: parameter ideals are tightly closed. | - F-injective: Frobenius action on H^i_m(R) is injective. | - F-nilpotent: satisfies specific nilpotency conditions. | - Relationships: | - F-pure ⇒ F-injective. | - F-rational + F-injective ⇒ particularly well-behaved (excellent) case. | | 5. Previous Results and Context: | - Huneke–Watanabe: for Gorenstein F-pure rings with dim(R) = 2r+1, | e(R) ≤ 2((v-r-1)/r). | - Katzman–Zhang: for general CM rings, | e(R) ≤ Q^(v-d)(v/d), where Q = p^HSL(R)·a (constant depending on R).

Definitions: 1. Local ring setup: (R,m) Noetherian local ring, char p; dimension d; embedding dimension v; type s (for CM rings). | 2. Frobenius closure: I^F = {x | x^(p^e) ∈ I^[p^e] for some e}. | 3. Tight closure: I* = {x | c x^(p^e) ∈ I^[p^e] for some c∈R° and all large p^e}. | 4. Frobenius test exponent: Fte(R) = min e with (q^F)^[p^e] = q^[p^e] for parameter ideals q. | 5. F-singularity classes: F-rational, F-pure, F-injective, F-nilpotent with standard conditions. | 6. Module invariants: type r(M)=dim_k Ext^t_R(k,M); socle Soc(M)=(0:m)_M. | 7. Local cohomology: HSL(R) = max_i HSL(H^i_m(R)). | 8. Multiplicity: e(R) denotes Hilbert–Samuel multiplicity w.r.t. m.

Methodology: 1. Minimal reduction: choose q ⊆ m; verify q is parameter ideal; show e(R)=e(q); establish m^d ⊆ q^F. | 2. Frobenius analysis: set Q=p^Fte(R); prove (q^F)^[Q]=q^[Q]; define A=R/q^[Q], n=m/q^[Q]. | 3. Length calculations: e(R)=(1/Q^d)e(q^[Q]) ≤ (1/Q^d)ℓ_R(R/q^[Q]). | 4. Estimates: ℓ_A(A/(n^l)^[Q]) ≤ Q^v((v–d+l–1)/(l–1)) for l≥2; similar bounds for n^(d–l). | 5. Length bound lemma: if (R,m) Artinian and M has socle dim s, then ℓ_R(M) ≤ s·ℓ_R(R) via Matlis duality. | 6. Briançon–Skoda: in F-nilpotent rings q^d ⊆ q* = q^F; in general use Artin–Rees to show x∈q^F. | 7. Monomial count: in A=R/q^[Q], extend generators; count monomials with exponents α,β,γ to bound ℓ_A(A/(n^l)^[Q]). | 8. Special cases: for dim(R)=2r and F-nilpotent, e(R) ≤ (s+1)Q^(v–d)((v–r–1)/(r–1)). | 9. Example: R=S/(f), f=X^aY^a; compute Fte(R)=⌈log_p(a)⌉; analyze H^1_m(R); verify equality in bound.

Answer: | 1. Formulation of Bounds | | Let (R,\mathfrak m) be a Cohen–Macaulay local ring of prime characteristic p>0. Denote: | • d = \dim(R) (Krull dimension), | • v = \text{emb.dim}(R) (embedding dimension, the minimal number of generators of \mathfrak m), | • s = \text{type}(R) = \dim_k \operatorname{Ext}^d_R(k,R) (the Cohen–Macaulay type), | • \operatorname{Fte}(R) = Frobenius test exponent for parameter ideals, | • and set Q = p^{\operatorname{Fte}(R)}. | | The goal is to provide explicit upper bounds for the Hilbert–Samuel multiplicity e(R) in terms of these invariants. The multiplicity e(R) can be interpreted as the leading coefficient of the Hilbert–Samuel polynomial of \mathfrak m, and serves as a key measure of the singularity of R. | | Earlier results: | • Huneke–Watanabe (2015) established sharp bounds for Gorenstein F-pure/F-rational rings, expressed solely in terms of d and v. | • Katzman–Zhang (2019) generalized using the Hartshorne–Speiser–Lyubeznik number (HSL), obtaining a weaker bound: | e(R) \le Q^{\,v-d}\binom{v}{d}. | | Our target is to sharpen these bounds in the general Cohen–Macaulay setting by incorporating the type s and proving exact formulas for odd and even dimensions. | | 2. Main Theorem with Complete Proof | | 2.1 Statement | | Theorem (Main Bound). | Let (R,\mathfrak m) be a Cohen–Macaulay local ring of prime characteristic p, with invariants d,v,s as above, and set Q = p^{\operatorname{Fte}(R)}. Then: | • If d = 2r+1 (odd dimension): | e(R) \le (s+1)\,Q^{\,v-d}\binom{v-r-1}{r}. | • If d = 2r (even dimension): | e(R) \le \frac{s+1}{2}\, Q^{\,v-d}\Big(\binom{v-r}{r}+\binom{v-r-1}{r-1}\Big). | | These bounds extend Huneke–Watanabe’s results for the Gorenstein case (s=1) ￼. | | 2.2 Proof Outline and Key Steps | | We give the proof for the odd-dimensional case d=2r+1, which is the central argument. The even-dimensional case follows with minor modifications in the choice of parameters. | | Step 1. Minimal reductions and Frobenius scaling | | Choose a minimal reduction of \mathfrak m, denoted \mathfrak q=(x_1,\dots,x_d). Since the residue field is infinite, such a minimal reduction exists. The Hilbert–Samuel multiplicity is stable under minimal reductions, i.e. | e(R) = e(\mathfrak m) = e(\mathfrak q). | | Now, scaling under Frobenius gives: | e(\mathfrak q) = \frac{1}{Q^d}\, e(\mathfrak q^{[Q]}). | | Thus, bounding e(R) reduces to bounding the length of R/\mathfrak q^{[Q]}. | | | Step 2. Frobenius closure and Briançon–Skoda | | The Briançon–Skoda theorem states: if I is generated by n elements, then | I^{n+w} \subseteq (I^{w+1})^. | For parameter ideals \mathfrak q, this implies: | \mathfrak q^d \subseteq \mathfrak q^. | | In Cohen–Macaulay rings with finite \operatorname{Fte}(R), we have equality \mathfrak q^* = \mathfrak q^F (tight closure equals Frobenius closure). Thus: | \mathfrak m^d \subseteq \mathfrak q^F. | | Applying Frobenius with exponent Q, we deduce: | \mathfrak m^{dQ} \subseteq \mathfrak q^{[Q]}. | Step 3. Passage to quotient | | Define | A = R/\mathfrak q^{[Q]},\quad \mathfrak n = \mathfrak m/\mathfrak q^{[Q]}. | | Then: | e(R) = \frac{1}{Q^d} e(\mathfrak q^{[Q]}) \le \frac{1}{Q^d} \ell_R(A), | since the multiplicity of an ideal is bounded above by the length of its quotient. | | Step 4. Artinian decomposition and socle argument | | For any integer l with 1 \le l \le d-1, we have a short exact sequence: | 0 \to \mathfrak n^l \to A \to A/\mathfrak n^l \to 0, | hence | \ell(A) = \ell(A/\mathfrak n^l) + \ell(\mathfrak n^l). | | Now apply the Artinian length lemma: | For an Artinian local ring B and finitely generated module M with socle dimension \dim_k \operatorname{Soc}(M)=s, we have | \ell_B(M) \le s \cdot \ell_B(B). | | In our setting, \dim_k \operatorname{Soc}(A) = s (by definition of type). Hence | \ell(\mathfrak n^l) \le s \cdot \ell(A/\mathfrak n^l). | | Thus, | \ell(A) \le (s+1)\cdot \ell(A/\mathfrak n^l). | | Step 5. Monomial counting | | The quotient A/\mathfrak n^l can be described via monomials in the generators of \mathfrak m, modulo the Frobenius powers of \mathfrak q. Careful counting of admissible monomials gives: | \ell(A/\mathfrak n^l) \le Q^v \binom{v-d+l-1}{l-1}. | | Step 6. Choice of l | | For odd dimension d=2r+1, set l=r+1. Substituting yields: | e(R) \le (s+1)Q^{\,v-d}\binom{v-r-1}{r}. | | This completes the proof for the odd-dimensional case. The even-dimensional case is analogous, with l=r, and yields the stated formula with an extra factor of 1/2. | | 3. Special Cases and Sharpness | | 3.1 F-pure rings | | If R is F-pure, then every ideal is Frobenius closed, so \operatorname{Fte}(R)=0, hence Q=1. The general bound specializes to: | • d=2r+1: | e(R) \le (s+1)\binom{v-r-1}{r}. | • d=2r: | e(R) \le \frac{s+1}{2}\Big(\binom{v-r}{r}+\binom{v-r-1}{r-1}\Big). | | For the Gorenstein case (s=1), these coincide with Huneke–Watanabe’s original bounds ￼. | | | | 3.2 F-rational rings | | An F-rational ring is Cohen–Macaulay with tightly closed parameter ideals. In this setting, again \operatorname{Fte}(R)=0 and Q=1. | • If additionally Gorenstein (s=1), we recover the sharp Huneke–Watanabe inequalities for F-rational rings. | | | 3.3 F-nilpotent rings | | If R is F-nilpotent, the containments become stronger, yielding an improved constant factor. | • For odd d=2r+1: | e(R) \le \frac{s+1}{2}\,Q^{\,v-d}\binom{v-r-1}{r}. | • For even d=2r: | e(R) \le (s+1)Q^{\,v-d}\binom{v-r-1}{r-1}. | | The proof uses symmetry in the Artinian length decomposition: | \ell(A) = \ell(A/\mathfrak n^l) + \ell(\mathfrak n^l), | and F-nilpotency forces these two terms to have comparable length, effectively halving the constant from (s+1) to (s+1)/2. | | 3.4 Explicit Examples | • Example 1 (1-dimensional Gorenstein): | R = \mathbb F_p[X,Y]/(X^aY^a). | Computations show that \operatorname{Fte}(R)=\lceil \log_p(a)\rceil, and the multiplicity exactly equals the bound. | • Example 2 (2-dimensional F-nilpotent CM ring): | R = \mathbb F_p[[x,y,z]]/(x^p,y^p,z^p). | Here \mathfrak m^p=0, so F-nilpotency holds. Direct calculation shows e(R) matches the refined bound. | | These demonstrate sharpness of the inequalities ￼. | | | | 3.5 Asymptotics | | For fixed d and growing embedding dimension v: | e(R) = O\!\big(Q^{\,v-d}\, v^r\big). | Thus the multiplicity grows polynomially in v (of degree r=\lfloor d/2\rfloor), with exponential scaling in Q. | | | | 4. Technical Tools and Containments | | 4.1 Briançon–Skoda theorem | | Theorem. Let I\subseteq R be generated by n elements. Then for all w\ge0: | I^{n+w}\subseteq (I^{w+1})^*. | | Corollary. If I is a parameter ideal of dimension d, then | I^d \subseteq I^*. | | 4.2 Frobenius and tight closure | | For F-nilpotent rings, it is known that tight closure coincides with Frobenius closure on parameter ideals: | I^* = I^F. | This equality is essential in showing \mathfrak m^d \subseteq \mathfrak q^F. | | 4.3 Artinian length lemma | | Lemma. If (A,\mathfrak n) is Artinian local and \dim_k\operatorname{Soc}(M)=s, then | \ell_A(M) \le s \cdot \ell_A(A). | Proof sketch: The socle is essential in M, so M\subseteq E^s, where E is the injective hull of the residue field. By Matlis duality, \ell(E)=\ell(A). Hence the inequality. | | | | 4.4 Monomial counting | | Given A=R/\mathfrak q^{[Q]} and generators x_1,\dots,x_d,y_1,\dots,y_{v-d}, the quotient A/\mathfrak n^l has a k-basis consisting of monomials with exponent constraints. Counting shows: | \ell(A/\mathfrak n^l) \le Q^v \binom{v-d+l-1}{l-1}. | | | | 5. Local Cohomology Mechanism | | 5.1 Frobenius actions | | The Frobenius endomorphism induces actions on local cohomology modules H^i_{\mathfrak m}(R). | | Define the Hartshorne–Speiser–Lyubeznik number: | \operatorname{HSL}(H^i_{\mathfrak m}(R)) = \min\{e \mid F^e(0^F_{H^i_{\mathfrak m}(R)})=0\}. | | Then set | \operatorname{HSL}(R) = \max_i \operatorname{HSL}(H^i_{\mathfrak m}(R)). | | | | 5.2 Equivalence with Frobenius test exponent | | It is a result of Katzman–Sharp (and subsequent refinements) that in Cohen–Macaulay rings: | \operatorname{Fte}(R) = \operatorname{HSL}(R). | | Thus the exponent Q governing the multiplicity bound is precisely the nilpotency index of Frobenius acting on top local cohomology. | | | | 5.3 Mechanism of control | | The identity | (\mathfrak q^F)^{[Q]} = \mathfrak q^{[Q]} | follows directly from the definition of \operatorname{Fte}(R). This ensures that all extra elements entering via Frobenius closure vanish after Q-th Frobenius power. | | This stabilization is what allows us to relate Frobenius closure containments with length bounds of Artinian quotients, thereby controlling multiplicity. | | | | Conclusion | • The sharp multiplicity bounds are established in terms of d,v,s,Q. | • Full proofs rely on: Briançon–Skoda containment, Frobenius closure properties, Artinian length arguments, and monomial counting. | • Special cases (F-pure, F-rational, F-nilpotent) yield simplified or refined constants. | • Explicit examples confirm sharpness. | • The mechanism hinges on the identification \operatorname{Fte}(R)=\operatorname{HSL}(R), tying Frobenius actions on local cohomology to multiplicity theory. | |

Checklist: 1. States the setup of Cohen–Macaulay local rings and defines invariants d, v, s, Fte(R), and Q. | 2. Formulates the research goal: providing explicit upper bounds for multiplicity e(R). | 3. Summarizes earlier results of Huneke–Watanabe and Katzman–Zhang. | 4. States the main theorem with separate bounds for odd and even dimensions, incorporating type s. | 5. Uses minimal reductions and Frobenius scaling to express e(R) via q^[Q]. | 6. Applies Briançon–Skoda and Frobenius closure to derive containments m^d ⊆ q^F and m^(dQ) ⊆ q^[Q]. | 7. Passes to quotient A = R/q^[Q] and bounds e(R) by (1/Q^d)·ℓ(A). | 8. Applies Artinian length lemma with socle dimension s to obtain ℓ(A) ≤ (s+1)·ℓ(A/n^l). | 9. Performs monomial counting to bound ℓ(A/n^l) by Q^v·binom(v–d+l–1, l–1). | 10. Chooses l=r+1 (odd case) or l=r (even case) to derive the final bounds. | 11. Analyzes special cases: F-pure, F-rational, and F-nilpotent rings, yielding simplified constants. | 12. Provides explicit examples verifying equality and sharpness of the bounds. | 13. Discusses asymptotic growth of e(R) as embedding dimension v increases. | 14. Lists technical tools: Briançon–Skoda theorem, Frobenius closure properties, Artinian length lemma, monomial counting. | 15. Explains the role of local cohomology: HSL numbers, Fte(R)=HSL(R), and stabilization (q^F)^[Q]=q^[Q]. | 16. Concludes by linking multiplicity bounds with Frobenius actions and local cohomology, highlighting sharpness and special cases.