López Garcia, Daniel The monodromy problem for hyperelliptic curves. (English) Zbl 1470.32001 Bull. Sci. Math. 170, Article ID 102998, 33 p. (2021). Summary: We study the Dynkin diagrams associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case \(y^4+g(x)\), which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, we solve the monodromy problem for direct sums of fourth degree polynomials. MSC: 32A08 Polynomials and rational functions of several complex variables Keywords:monodromy of direct sums of polynomials; Dynkin diagrams PDFBibTeX XMLCite \textit{D. López Garcia}, Bull. Sci. Math. 170, Article ID 102998, 33 p. (2021; Zbl 1470.32001) Full Text: DOI arXiv References: [1] A’Campo, N., Le groupe de monodromie du déploiement des singularités isolées de courbes planes I, Math. Ann., 1-32 (1975) · Zbl 0316.14011 [2] Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S., Singularities of Differentiable Maps: Volume II Monodromy and Asymptotic Integrals, vol. 83 (1988), Springer Science & Business Media [3] Cerveau, D.; Neto, A. L., Irreducible components of the space of holomorphic foliations of degree two in \(\mathbb{CP}(n), n \geq 3\), Ann. Math., 577-612 (1996) · Zbl 0855.32015 [4] Christopher, C.; Mardešić, P., The monodromy problem and the tangential center problem, Funct. Anal. Appl., 44, 1, 22-35 (2010) · Zbl 1271.34035 [5] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (2013), Springer Science & Business Media [6] Da Fonseca, C.; Kowalenko, V., Eigenpairs of a family of tridiagonal matrices: three decades later, Acta Math. Hung., 1-14 (2019) · Zbl 1449.15012 [7] Doran, C.; Morgan, J., Mirror symmetry and integral variations of Hodge structure underlying one parameter families of Calabi-Yau threefolds, (V, AMS/IP Studies in Advanced Mathematics, vol. 38 (2006)) · Zbl 1116.14005 [8] Dulac, H., Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre, vol. 32 (1908), Gauthier-Villars · JFM 39.0374.01 [9] Françoise, J. P., Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergod. Theory Dyn. Syst., 16, 1, 87-96 (1996) · Zbl 0852.34008 [10] Gavrilov, L., Petrov modules and zeros of Abelian integrals, Bull. Sci. Math., 122, 8, 571-584 (1998) · Zbl 0964.32022 [11] Gavrilov, L.; Movasati, H., The infinitesimal 16th Hilbert problem in dimension zero, Bull. Sci. Math., 131 (2007) · Zbl 1119.14040 [12] Ilyashenko, Y., The origin of limit cycles under perturbation of the equation \(d w / d z = - r_z / r_w\), where \(r(z, w)\) is a polynomial, Mat. Sb., 120, 3, 360-373 (1969) [13] Lamotke, K., The topology of complex projective varieties after S. Lefschetz, Topology, 20, 1, 15-51 (1981) · Zbl 0445.14010 [14] López Garcia, D., Homology supported in Lagrangian submanifolds in mirror quintic threefolds, Can. Math. Bull. (2020) [15] Losonczi, L., Eigenvalues and eigenvectors of some tridiagonal matrices, Acta Math. Hung., 60, 3-4, 309-322 (1992) · Zbl 0771.15004 [16] Movasati, H., Abelian integrals in holomorphic foliations, Rev. Mat. Iberoam., 20, 1, 183-204 (2004) · Zbl 1055.37057 [17] Movasati, H., Center conditions: rigidity of logarithmic differential equations, J. Differ. Equ., 197, 1, 197-217 (2004) · Zbl 1049.32033 [18] Movasati, H., A Course in Hodge Theory, with Emphasis on Multiple Integrals (2017), IP: IP Boston, To be published by [19] Neto, A. L., Componentes irredutíveis dos espaços de folheações, Publicaçoes Matematicas do IMPA (2007) · Zbl 1143.32019 [20] Neto, A. L., Foliations with a Morse center, J. Singul., 9, 82-100 (2014) · Zbl 1333.37064 [21] Roussarie, R., Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, vol. 164 (1998), Birkhäuser · Zbl 0898.58039 [22] Yueh, W., Eigenvalues of several tridiagonal matrices, Appl. Math. E-Notes, 5, 66-74, 210-230 (2005) [23] Zare, Y., Center conditions: pull back of differential equations, Trans. Am. Math. Soc. (2017) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.