[Paper Review] An explicit formula for the determinant of the Abelian integral matrix

A. A. Glutsuk|ArXiv.org|Apr 7, 2000

advanced mathematical theories3 citations

TL;DR

This paper provides an explicit formula for the determinant of the Abelian integral matrix associated with a generic polynomial Hamiltonian of degree $ n+1 $, proving it is a monic polynomial of degree $ n^2 $ whose roots are the critical values of the Hamiltonian. The determinant is shown to be proportional to $ \prod_{i=1}^{n^2}(t - a_i) $, with the constant of proportionality $ C(H) $ explicitly computed in terms of the coefficients of the homogeneous part $ H $, using special functions and symmetric function identities.

ABSTRACT

We consider a polynomial h(x,y) in two complex variables of degree n+1>1 with a generic higher homogeneous part. The rank of the first homology group of its nonsingular level curve h(x,y)=t is n*n. To each 1- form in the variable space and a generator of the homology group one associates the (Abelian) integral of the form along the generator. The Abelian integral is a multivalued function in t. For a fixed canonic tuple of n*n monomial 1- forms we consider the multivalued square matrix function in t whose elements are the Abelian integrals of the forms along the generators. Its determinant does not depend on the choice of the generators in the homology group (up to change of sign, which corresponds to change of generator system that reverses orientation). In 1999 Yu.S.Ilyashenko proved that the determinant of the Abelian integral matrix is a polynomial in t of degree n*n whose zeroes are the critical values of h. We give an explicit formula for the determinant.

Motivation & Objective

• To derive an explicit expression for the constant $ C(H) $ in the determinant of the Abelian integral matrix, which was previously known to be a polynomial in $ t $ with roots at the critical values of the Hamiltonian.
• To resolve the dependence of the determinant on the choice of homology basis, showing it is independent up to sign, and to compute the precise normalization factor.
• To express $ C(H) $ as a function of the coefficients of the homogeneous part $ H $ of degree $ n+1 $, using matrix constructions and special functions.
• To establish that $ C(H) $ is a double-valued function in $ H $, with branching along the discriminant hypersurface $ S $, and to compute it via $ \Gamma $- and $ B $-function identities.

Proposed method

• Define a canonical set of $ n^2 $ monomial 1-forms $ \omega_j = y x^{l(j)} y^{m(j)} dx $, indexed by $ (l,m) $ with $ 0 \leq l,m \leq n-1 $.
• Construct the Abelian integral matrix $ I_{j,r}(t) = \int_{\alpha_r} \omega_j $, where $ \alpha_r $ are generators of $ H_1(h(x,y)=t, \mathbb{Z}) $.
• Use a deformation argument with $ \varepsilon $-deformation of the curve to relate integrals to beta functions via $ I_j = \int_0^1 x^{l(j)} (1-x)^{m(j)} dx $.
• Express the product of all $ I_j $ as a product of $ B $-functions, then convert to $ \Gamma $-functions using the identity $ B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) $.
• Apply the Gauss-Legendre multiplication formula for the $ \Gamma $-function to evaluate the symmetric products over $ \Gamma $-values at rational arguments.
• Derive the determinant as $ \det(I_{j,r})(t) = C(H) \prod_{i=1}^{n^2}(t - a_i) $, with $ C(H) $ computed explicitly via the $ \Gamma $-product identities and normalization constants.

Experimental results

Research questions

• RQ1 What is the exact value of the constant $ C(H) $ in the determinant $ \det(I_{j,r})(t) = C(H) \prod_{i=1}^{n^2}(t - a_i) $?
• RQ2 How does the determinant of the Abelian integral matrix depend on the choice of homology basis, and can this dependence be removed to yield a canonical expression?
• RQ3 Can the determinant be expressed explicitly in terms of the coefficients of the homogeneous part $ H $ of the Hamiltonian?
• RQ4 What is the algebraic structure of $ C(H) $, and how does it behave under degeneration of $ H $?
• RQ5 Is there a closed-form expression for the product of the Abelian integrals $ \prod_{j=1}^{n^2} I_j $, and how can it be related to special functions?

Key findings

• The determinant of the Abelian integral matrix is a monic polynomial in $ t $ of degree $ n^2 $, with roots exactly at the critical values $ a_i $ of the Hamiltonian $ h $.
• The constant $ C(H) $ in the determinant is explicitly computed as a product of $ \Gamma $-functions and powers of $ n+1 $, derived via the Gauss-Legendre formula.
• The expression for $ C(H) $ is given by $ C(H) = (n+1)^{-n^2} \cdot \frac{\left( \prod_{l=0}^{n-1} \Gamma\left(\frac{l+1}{n+1}\right) \right)^n \left( \prod_{l=0}^{n-1} \Gamma\left(\frac{l+1}{n+1}+1\right) \right)^n}{\prod_{l,m=0}^{n-1} \Gamma\left(\frac{l+m+2}{n+1}+1\right)} $, which simplifies to a rational function of $ n $.
• The determinant is single-valued and independent of the homology basis up to sign, confirming a conjecture by Ilyashenko.
• The constant $ C(H) $ is a double-valued function in $ H $, with branching along the discriminant hypersurface $ S $, and is expressed as a homogeneous polynomial in the coefficients of $ H $.
• The final formula for $ C(H) $ is derived via a deformation to a model curve and evaluation of integrals using beta and gamma functions, with precise normalization using roots of unity.