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Monodromies of singularities of the Hadamard and eñe product

Ricardo Pérez-Marco

TL;DR

The paper establishes explicit holomorphic monodromy formulas for the Hadamard and exponential eñe products, showing how singularities with holomorphic monodromy are preserved and how their monodromies combine via convolution-type expressions. It develops integrable/totally holomorphic frameworks, extends the results from single to higher multiplicities, and provides general Hadamard and eñe formulas with residue and integral terms. The work yields Borel-type invariance results for holomorphic monodromy, introduces the polynomial logarithmic monodromy (PLM) class, and proves a divisor interpretation of the eñe product, connecting zeros/poles to monodromy data within the big Witt ring perspective. Applications include the monodromy of polylogarithms and a divisor-based viewpoint for the eñe product, illustrating the interaction between analytic continuation, algebraic structures, and divisor theory.

Abstract

We prove that singularities with holomorphic monodromies are preserved by the Hadamard product. We find an explicit formula for the monodromy of the singularities, and similar formulas for the exponential eñe product. Using these formulas we get new direct proofs of classical results and the invariance of some rings of functions by Hadamard and eñe product (which is the product of the big Witt ring). This explains the good behavior of the eñe product on divisors.

Monodromies of singularities of the Hadamard and eñe product

TL;DR

The paper establishes explicit holomorphic monodromy formulas for the Hadamard and exponential eñe products, showing how singularities with holomorphic monodromy are preserved and how their monodromies combine via convolution-type expressions. It develops integrable/totally holomorphic frameworks, extends the results from single to higher multiplicities, and provides general Hadamard and eñe formulas with residue and integral terms. The work yields Borel-type invariance results for holomorphic monodromy, introduces the polynomial logarithmic monodromy (PLM) class, and proves a divisor interpretation of the eñe product, connecting zeros/poles to monodromy data within the big Witt ring perspective. Applications include the monodromy of polylogarithms and a divisor-based viewpoint for the eñe product, illustrating the interaction between analytic continuation, algebraic structures, and divisor theory.

Abstract

We prove that singularities with holomorphic monodromies are preserved by the Hadamard product. We find an explicit formula for the monodromy of the singularities, and similar formulas for the exponential eñe product. Using these formulas we get new direct proofs of classical results and the invariance of some rings of functions by Hadamard and eñe product (which is the product of the big Witt ring). This explains the good behavior of the eñe product on divisors.

Paper Structure

This paper contains 15 sections, 35 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction and background.
  2. Hadamard and eñe product.
  3. Holomorphic monodromy formulas.
  4. Convolution formulas.
  5. Monodromy of singularities and monodromy operator.
  6. Monodromy formula for integrable singularities.
  7. Proof of the totally holomorphic monodromy formula: Single singularity case
  8. Proof for higher multiplicity singularities.
  9. General holomorphic monodromy formula.
  10. Formula for the Hadamard product.
  11. Formula for the exponential eñe product.
  12. Applications.
  13. Application 1: Monodromy of polylogarithms.
  14. Application 2: Polynomial logarithmic monodromy class.
  15. Application 3: Divisor interpretation of the eñe product.

Key Result

Proposition 1.3

When $\Delta_{\alpha}^2 F =0$, we can decompose $F$ uniquely as where $F_0$ has a uniform singularity at $0$.

Figures (2)

  • Figure 1: Choreographic monodromy integration contour.
  • Figure 2: Homotopical deformation of the integration path when $z_0$ turns around $1$.

Theorems & Definitions (61)

  • Definition 1.1: Monodromy of an isolated singularity
  • Definition 1.2: Holomorphic monodromy
  • Proposition 1.3
  • Definition 1.4: Totally holomorphic singularity
  • Theorem 1.5: Holomorphic monodromy formula for the Hadamard product
  • Corollary 1.6
  • Proposition 1.7
  • Theorem 1.8: Holomorphic monodromy formula for the exponential eñe product
  • Corollary 1.9
  • Corollary 1.10
  • ...and 51 more