[Paper Review] Algebraicity of some Hilbert-Kunz multiplicities (modulo a conjecture)

Paul Monsky|ArXiv.org|Jul 15, 2009

Coding theory and cryptography5 references3 citations

TL;DR

Assuming a precise conjecture on colengths of ideals in characteristic 2, this paper establishes the algebraicity of Hilbert-Kunz multiplicities and series for certain power series, including $ H = uv + h $ and $ H = g(u,v) + h $, by employing infinite matrix theory and Grothendieck group techniques. The key result is that the Hilbert-Kunz multiplicity of $ h = x^3 + y^3 + xyz $ is algebraic: $ \frac{4}{3} + \frac{5}{14\sqrt{7}} $, and the Hilbert-Kunz series lies in a degree-2 algebraic extension of $ \mathbb{Q}(w) $.

ABSTRACT

Let F be a finite field of characteristic 2 and h be the element x^3+y^3+xyz of F[[x,y,z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x^q,y^q,z^q,h^j) for q a power of 2. We also showed that if the conjecture holds then the Hilbert-Kunz series of H=uv+h is algebraic (of degree 2) over Q(w), and that mu(h) is algebraic (explicitly, (4/3)+(5/14)sqrt(7)). In this note, assuming the same conjecture, we use a theory of infinite matrices to rederive this result, and we extend it to a wider class of H; for example H=g(u,v)+h. In a follow-up paper, under the same hypothesis, we will show that transcendental Hilbert-Kunz multiplicities exist.

Motivation & Objective

• To establish the algebraicity of Hilbert-Kunz multiplicities and series for specific power series in characteristic 2, assuming a conjecture on ideal colengths.
• To extend the algebraicity result from $ H = uv + h $ to a broader class of $ H = g(u,v) + h $, where $ g $ is a homogeneous polynomial.
• To use infinite matrix theory and Grothendieck group structures to analyze the Hilbert-Kunz series and multiplicity.
• To show that the Hilbert-Kunz multiplicity of $ h = x^3 + y^3 + xyz $ is algebraic, explicitly $ \frac{4}{3} + \frac{5}{14\sqrt{7}} $, under the conjecture.
• To demonstrate that the Hilbert-Kunz series lies in a splitting field of a certain polynomial, implying algebraicity over $ \mathbb{Q}(w) $.

Proposed method

• Introduces a bilinear product $ \# $ on the space $ X $ of functions $ [0,1] \cap \mathbb{Z}[1/2] \to \mathbb{Q} $, defined recursively using dyadic scaling.
• Defines $ \phi_f \in X $ as the Hilbert-Kunz function for a power series $ f $, encoding the colengths of ideals $ (u^q, \dots, f^i) $.
• Uses the limit $ \mu(\alpha) = \lim_{n \to \infty} \alpha(2^{-n}) \cdot 2^n $ to define the Hilbert-Kunz multiplicity for $ \alpha \in X $.
• Constructs a map $ \mathcal{L}_n(\alpha) $ from $ \alpha \in X $ to the Grothendieck group $ \Gamma_{\mathbb{Q}} $, using alternating sums of dimensions of graded pieces.
• Applies the theory of infinite matrices and the action of $ T_0, T_1 $ on modules to derive a recurrence for the Hilbert-Kunz series.
• Relies on the splitting field of a determinant $ \Psi_{T_0,T_1}(x,w) $ to show that the Hilbert-Kunz series lies in a finite extension of $ \mathbb{Q}(w) $, implying algebraicity.

Experimental results

Research questions

• RQ1 Under the conjecture on colengths of ideals $ (x^q, y^q, z^q, h^j) $, is the Hilbert-Kunz series of $ H = uv + h $ algebraic over $ \mathbb{Q}(w) $?
• RQ2 Can the algebraicity of the Hilbert-Kunz series be extended to $ H = g(u,v) + h $ for homogeneous $ g $?
• RQ3 Is the Hilbert-Kunz multiplicity of $ h = x^3 + y^3 + xyz $ algebraic, and if so, what is its explicit value?
• RQ4 What is the Galois-theoretic structure of the splitting field containing the Hilbert-Kunz series for $ H = u^6 + u^3v^3 + v^6 + h $?
• RQ5 Does the Hilbert-Kunz multiplicity generate the full degree-8 extension of $ \mathbb{Q} $, or a proper subfield, in the residue field at $ (1 - 16w) $?

Key findings

• Assuming the conjecture, the Hilbert-Kunz multiplicity of $ h = x^3 + y^3 + xyz $ is algebraic and explicitly $ \frac{4}{3} + \frac{5}{14\sqrt{7}} $.
• The Hilbert-Kunz series of $ H = uv + h $ is algebraic of degree 2 over $ \mathbb{Q}(w) $, as shown via the infinite matrix method.
• For $ H = u^6 + u^3v^3 + v^6 + h $, the Hilbert-Kunz series lies in a degree-8 extension of $ \mathbb{Q}(w) $, specifically $ \mathbb{Q}(w, u_1, u_2) $, where $ u_1^2 \in \mathbb{Q}(w) $ and $ u_2 $ has degree 4 over $ \mathbb{Q}(w) $.
• The residue class field at the prime $ (1 - 16w) $ in the integral closure of $ \mathbb{Q}[w] $ is a degree-8 extension of $ \mathbb{Q} $, and the Hilbert-Kunz multiplicity lies in this field.
• The Galois group of the splitting field of the relevant polynomial $ \Psi^* $ has order 48 and stabilizes the sets $ \{\rho, \rho^{-1}\}, \{\sigma, \sigma^{-1}\}, \{\tau, \tau^{-1}\} $.
• The Hilbert-Kunz multiplicity is believed to generate the full degree-8 extension of $ \mathbb{Q} $, though verifying this requires a computationally intensive calculation.