[Paper Review] Integrable derivations in the sense of Hasse-Schmidt for some binomial plane curve

María de la Paz Tirado Hernández|arXiv (Cornell University)|Jul 27, 2018

Advanced Differential Equations and Dynamical Systems2 references3 citations

TL;DR

This paper characterizes the module of integrable Hasse-Schmidt derivations for the quotient ring $ k[x,y]/ olimits\langle x^n - y^q \rangle $ in positive characteristic, establishing conditions under which derivations are $ m $-integrable for $ m \geq 1 $ and $ m = \infty $. It shows that integrability depends on divisibility of $ n $ and $ q $ by the characteristic $ p $, and provides explicit generators for the integrable derivation modules in various cases, including examples with $ h = x^4 + y^6 + y^7 $ and $ h = x^3 + y^5 + x^2y^2 $ in characteristic 3.

ABSTRACT

We describe the module of integrable derivations in the sense of Hasse-Schmidt of the quotient of the polinomial ring in two variables over an ideal generated by the equation x^n-y^q.

Motivation & Objective

• To describe the module of $ m $-integrable Hasse-Schmidt derivations for the ring $ A = k[x,y]/\langle x^n - y^q \rangle $, where $ k $ is a reduced ring of positive characteristic $ p $.
• To analyze the integrability of derivations in the sense of Hasse-Schmidt when $ n $ or $ q $ is divisible by $ p $, using a relationship between $ \langle f \rangle $ and $ \langle f^p \rangle $.
• To compute explicit generators for the module of integrable derivations in specific examples, including $ h = x^4 + y^6 + y^7 $ and $ h = x^3 + y^5 + x^2y^2 $ in characteristic 3.
• To establish conditions under which derivations are $ \infty $-integrable, particularly focusing on logarithmic integrability with respect to the defining ideal.

Proposed method

• Uses the correspondence between Hasse-Schmidt derivations and $ k $-algebra homomorphisms $ \varphi: A \to A[|\mu|]_m $, where $ \varphi(a) = \sum D_i(a)\mu^i $.
• Applies truncation maps $ \tau_{nm} $ to relate $ m $-integrable and $ n $-integrable derivations for $ n > m $.
• Employs logarithmic derivations with respect to the ideal $ \langle h \rangle $, requiring $ D_i(h) \in \langle h \rangle $ for all $ i $.
• Relies on inductive lemmas (e.g., Lemma 3.1 and Lemma 3.2) to construct higher-order components $ D_i $ of the derivation sequence in $ \operatorname{HS}_k(A;m) $.
• Uses the structure of the polynomial ring and quotient relations to reduce integrability conditions to ideal membership in $ \langle h \rangle $.
• Analyzes the coefficient of $ \mu^i $ in $ \varphi(h) $ to determine whether $ D_i $ can be extended consistently while preserving integrability.

Experimental results

Research questions

• RQ1 When is a $ k $-derivation on $ k[x,y]/\langle x^n - y^q \rangle $ $ m $-integrable for $ m \geq 1 $ in positive characteristic?
• RQ2 How does the integrability of derivations change when $ n $ or $ q $ is divisible by the characteristic $ p $?
• RQ3 What are the explicit generators of the module $ \operatorname{IDer}_k(A;m) $ for $ A = k[x,y]/\langle x^n - y^q \rangle $ in specific cases?
• RQ4 Under what conditions is a derivation $ \infty $-integrable, particularly in the logarithmic sense with respect to $ \langle h \rangle $?
• RQ5 How do the integrability conditions for $ \varphi(h) \in \langle h \rangle $ constrain the choice of higher-order components $ D_i $?

Key findings

• For $ h = x^4 + y^6 + y^7 $ in characteristic $ p \neq 2 $, the module $ \operatorname{IDer}_k(A;i) $ is generated by $ \overline{\partial_x} $ for $ 1 \leq i < 4 $, by $ \overline{x\partial_x}, \overline{y^2\partial_x} $ for $ 4 \leq i < 8 $, and by $ \overline{x^2\partial_x}, \overline{xy\partial_x}, \overline{y^2\partial_x} $ for $ i \geq 8 $.
• The derivation $ y^2\partial_x $ is $ h $-logarithmically $ \infty $-integrable, as shown by constructing $ D_i $ with $ v_{4i} \in \langle y^2 \rangle $ satisfying the integrability condition.
• The derivation $ x^2\partial_x $ is $ h $-logarithmically integrable because a solution exists with $ v_4 \in \langle x^4, h \rangle $, and Lemma 3.2 ensures inductive extension.
• For $ h = x^3 + y^5 + x^2y^2 $ in characteristic 3, the module $ \operatorname{IDer}_k(A) $ is generated by $ \overline{\delta_1} = \overline{x^2\partial_x + y^3\partial_y} $ and $ \overline{\delta_2} = \overline{2y^2\partial_x + (x + y^2)\partial_y} $.
• The integrability of $ \delta_1 $ is established via Lemma 3.3, which ensures existence of $ u_i \in \langle x^2 \rangle $ such that the $ \mu^i $-coefficient of $ \varphi(h) $ lies in $ \langle h \rangle $.
• The integrability of $ \delta_2 $ is confirmed by Lemma 3.4, which constructs $ u_i \in \langle xy, y^3 \rangle $ and $ v_i \in \langle y^2 \rangle $ satisfying the required ideal membership for the $ \mu^i $-coefficient.