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2. The Hermitian two matrix model with an even quartic potential
- Author
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Arno B. J. Kuijlaars, Maurice Duits, and Man Yue Mo
- Subjects
Polynomial ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,Mathematical analysis ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,30E25, 60B20 ,Measure (mathematics) ,Hermitian matrix ,Matrix (mathematics) ,symbols.namesake ,Quartic function ,FOS: Mathematics ,Method of steepest descent ,symbols ,Riemann–Hilbert problem ,Complex Variables (math.CV) ,Mathematical Physics ,Mathematics - Probability ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider the two matrix model with an even quartic potential W(y)=y^4/4+alpha y^2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M_1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M_1. Our results generalize earlier results for the case alpha=0, where the external field on the third measure was not present., 123 pages, 15 figures
- Published
- 2012
3. Quasi-ordinary power series and their zeta functions
- Author
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E. Artal Bartolo, A. Melle Hernández, Ignacio Luengo, and Pi. Cassou-Noguès
- Subjects
Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebraically closed field ,Algebraic Geometry (math.AG) ,14B05, 14E15, 32S50 ,Mathematics ,Polynomial (hyperelastic model) ,Mathematics - Number Theory ,Applied Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Order (ring theory) ,Algebraic number field ,Igusa zeta-function ,Riemann zeta function ,Monodromy ,symbols ,010307 mathematical physics - Abstract
The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h,T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h,T)=P(T)/Q(T)$ such that almost all the candidate poles given by $Q(T)$ are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on $h^{-1}(0).$ In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if $h$ is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function., Comment: 74 pages
- Published
- 2005
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