[Paper Review] Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves

Yutaka Matsui, Kiyoshi Takeuchi|ArXiv.org|Sep 18, 2008

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper generalizes Libgober-Sperber's formula for monodromy zeta functions at infinity using sheaf-theoretic methods, particularly constructible sheaves and nearby cycle functors. It provides explicit formulas for global monodromy zeta functions along fibers over bifurcation points of polynomial maps, including the surprising result that the constant term of a non-convenient polynomial is typically a bifurcation point. The results are fully described by Newton polyhedra associated with the polynomials, extending to multi-variable polynomial maps and complete intersection fibers.

ABSTRACT

By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained.

Motivation & Objective

• To generalize Libgober-Sperber's formula for monodromy zeta functions at infinity to non-convenient polynomials using sheaf-theoretic methods.
• To derive explicit formulas for the global monodromy zeta functions along fibers over bifurcation points of polynomial maps, including the constant term as a bifurcation point in general.
• To extend the theory to polynomial maps $ f: \mathbb{C}^n \to \mathbb{C}^k $ with $ k \geq 1 $, particularly for complete intersection fibers.
• To express all results in terms of Newton polyhedra and toric compactifications, eliminating reliance on meromorphic extensions with indeterminate points.

Proposed method

• Use of constructible sheaves and nearby cycle functors to provide a functorial, sheaf-theoretic proof of Libgober-Sperber's formula.
• Blowing up the meromorphic extension $ \widetilde{f} $ of $ f $ to resolve indeterminacies and compactify $ (\mathbb{C}^*)^n $ via toric geometry.
• Definition of the global monodromy zeta function $ \zeta_f^b(t) $ via restriction to small loops around bifurcation points $ b \in B_f $.
• Computation of zeta functions via Euler integrals of local monodromy zeta functions over strata $ T_S \cap W \cap \{f_k(x) - c = 0\} $.
• Introduction of the $ k $-th principal monodromy zeta function $ \zeta_{f,k}^c(t) $ for maps $ f: \mathbb{C}^n \to \mathbb{C}^k $, defined via Newton polyhedra $ \Gamma_\infty^S(f_k) $.
• Use of strict non-degeneracy conditions and the decomposition of the zeta function over subsets $ S \subset \{1,\dots,n\} $ with $ m(S) \leq \sharp S $.

Experimental results

Research questions

• RQ1 Can Libgober-Sperber's formula for monodromy zeta functions at infinity be generalized to non-convenient polynomials using sheaf-theoretic methods?
• RQ2 Is the constant term of a non-convenient polynomial always a bifurcation point of the polynomial map?
• RQ3 Can explicit formulas for the global monodromy zeta function $ \zeta_f^b(t) $ along fibers over bifurcation points $ b \in B_f $ be derived directly from Newton polyhedra?
• RQ4 How can the monodromy zeta functions be generalized to multi-variable polynomial maps $ f: \mathbb{C}^n \to \mathbb{C}^k $ with complete intersection fibers?
• RQ5 What is the role of the Newton polyhedron $ \Gamma_\infty^S(f_k) $ in decomposing the $ k $-th principal monodromy zeta function $ \zeta_{f,k}^c(t) $?

Key findings

• The constant term $ a $ of a non-convenient polynomial $ f(x) = \sum_{v \in \mathbb{Z}_{\geq 0}^n} a_v x^v $ is a bifurcation point of $ f $ in general, a result not previously known.
• A new sheaf-theoretic proof of Libgober-Sperber's formula is given, resolving indeterminacies in the meromorphic extension $ \widetilde{f} $ via toric compactification.
• For any bifurcation point $ b \in B_f $, the global monodromy zeta function $ \zeta_f^b(t) $ is explicitly computed using Newton polyhedra and constructible sheaves.
• The jumping number $ \chi(f^{-1}(a)) - \chi(f^{-1}(c)) \in \mathbb{Z} $ is expressed in terms of the Newton polyhedron of $ f - a $, as shown in Corollaries 4.5 and 4.9.
• For polynomial maps $ f: \mathbb{C}^n \to \mathbb{C}^k $, the $ k $-th principal monodromy zeta function $ \zeta_{f,k}^c(t) $ is decomposed over subsets $ S \subset \{1,\dots,n\} $ with $ \Gamma_\infty^S(f_k) \supsetneq \{0\} $ and $ m(S) \leq \sharp S $.
• When $ f $ is strictly non-degenerate and $ m(S) = k $ for all $ S $ with $ \Gamma_\infty^S(f_k) \supsetneq \{0\} $, the zeta function simplifies to $ \zeta_{f,k}^c(t) = \widetilde{\zeta_{f,k}^c}(t) $, where $ \widetilde{\zeta_{f,k}^c}(t) $ is the Euler integral of the local monodromy zeta function.