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On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

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Abstract

In Cassaigne and Maillot (J Number Theory 83:226–255, 2000) and, later on, in Akatsuka (J Number Theory 129:2713–2734, 2009) the authors introduced zeta Mahler measure functions for multivariate polynomials [Cassaigne and Maillot (J Number Theory 83:226–255, 2000) called them “zeta Igusa” functions, but we follow here the terminology of Akatsuka (J Number Theory 129:2713–2734, 2009)]. We generalize this notion by defining a zeta Mahler measure function \({\mathcal {Z}}_X(\cdot ,f):{\mathbb {C}}\longrightarrow {\mathbb {C}}\), where X is a compact probability space and \(f:X\longrightarrow {\mathbb {C}}\) is a function bounded almost everywhere in X. We give sufficient conditions that imply that this function is holomorphic in certain domains. Zeta Mahler measure functions contains big amounts of information about the expected behavior of f on X. This generalization is motivated by the study of several quantities related to numerical methods that solve systems of multivariate polynomial equations. We study the functions \({\mathcal {Z}}(\cdot ,{1}/{\Vert \cdot \Vert _\mathrm{aff}})\), \({\mathcal {Z}}(\cdot ,{1}/{\mu _\mathrm{norm}})\) and \({\mathcal {Z}}(\cdot ,\mathrm JAC)\), respectively associated to the norm of the affine zeros (\(\Vert \cdot \Vert _\mathrm{aff}\)), the non-linear condition number (\(\mu _\mathrm{norm}\)) and the Jacobian determinant (JAC) of complete intersection zero-dimensional projective varieties. We find the exact value of these functions in terms of Gamma functions and we also describe their respective domains of holomorphy in \({\mathbb {C}}\). With the exact value of these zeta functions we can immediately prove and exhibit expectations of some average properties of zero-dimensional algebraic varieties. For instance, the exact knowledge of \({\mathcal {Z}}(\cdot ,{1}/{\Vert \cdot \Vert _\mathrm{aff}})\) yields as a consequence that the expectation of the mean of the logarithm of the norms of the affine zeros of a random system of polynomial equations is one half of the nth harmonic number \(H_n\). Other conclusions are exhibited along the manuscript. Using these generalized zeta functions we exhibit the exact value of the arithmetic height of the hyper-surface known as the discriminant variety (roughly speaking the variety formed by all systems of equations having a singular zero).

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  1. Depto. de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, 39071, Santander, Spain

    Luis M. Pardo & Mario Pardo

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  1. Luis M. Pardo
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  2. Mario Pardo
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Correspondence to Luis M. Pardo.

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Luis M. Pardo: Research was partially supported by Spanish Ministry of Science Grant MTM2014-55262-P.

Mario Pardo: Research was partially supported by Spanish Ministry of Science Grant MTM2014-55262-P and the Program of Pre-doctoral Grants of the University of Cantabria.

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Pardo, L.M., Pardo, M. On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant. AAECC 27, 303–358 (2016). https://doi.org/10.1007/s00200-016-0284-9

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