Search results for " 14D05"

showing 3 items of 3 documents

Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials

2014

It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.

Abelian integralPure mathematicsLogarithmApplied Mathematics34M35 34C08 14D05General Physics and AstronomyStatistical and Nonlinear PhysicsMorse codelaw.inventionPontryagin's minimum principlesymbols.namesakeMonodromylawPoincaré conjecturesymbolsPoint at infinitySpecial caseMathematics - Dynamical SystemsMathematical PhysicsMathematics
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Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

2003

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres ha…

Algebra and Number TheoryDegree (graph theory)Mathematics - Complex Variables14D0514J3214D07Complex multiplicationYukawa potentialRigidity (psychology)14J70ModuliCombinatoricsAlgebraMathematics - Algebraic Geometry14J70; 14D05; 14D07; 14J32HypersurfaceMathematics::Algebraic GeometryMathematikFOS: MathematicsGeometry and TopologyComplex Variables (math.CV)Algebraic Geometry (math.AG)Stack (mathematics)Mathematics
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Redundant Picard–Fuchs System for Abelian Integrals

2001

We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard-Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were…

MonomialPure mathematicsDynamical systems theoryDifferential equationDynamical Systems (math.DS)symbols.namesakeFOS: MathematicsMathematics - Dynamical SystemsAbelian groupComplex Variables (math.CV)Complex quadratic polynomialMathematicsDiscrete mathematicsMathematics - Complex Variables14D0514K20Applied Mathematics32S4034C0834C07symbolsEquivariant mapLocus (mathematics)Hamiltonian (quantum mechanics)32S2034C07; 34C08; 32S40; 14D05; 14K20; 32S20AnalysisJournal of Differential Equations
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