Abstract
We introduce new complex analytic integral transforms, the Lisbon Integrals, which naturally arise in the study of the affine space \(\mathbb {C}^k\) of unitary polynomials \(P_s(z)\) where \(s\in \mathbb {C}^k\) and \(z\in \mathbb {C}\), \(s_i\) identified to the i-th symmetric function of the roots of \(P_s(z)\). We completely determine the \(\mathscr {D}\)-modules (or systems of partial differential equations) the Lisbon Integrals satisfy and prove that they are their unique global solutions. If we specify a holomorphic function f in the z-variable, our construction induces an integral transform which associates a regular holonomic module quotient of the sub-holonomic module we computed. We illustrate this correspondence in the case of a 1-parameter family of exponentials \(f_t(z) = exp(t z)\) with t a complex parameter.
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Notes
despite the “denominators” in the formula
$$\begin{aligned} Trace(z^pf(z)ds_1 \wedge \cdots \wedge ds_{k-1} \wedge dz) = \left( \sum _{P_{s}(z_j)= 0} \frac{{z_j}^pf(z_j)}{P'_{s}(z_j)}\right) ds_1\wedge \cdots \wedge ds_k \end{aligned}$$theses forms have no singularity on the discriminant hypersurface \(\{\Delta (s) = 0\}\) in \(\mathbb {C}^k\).
We shall explain in the proof what we mean here.
Note that z and \(\partial _{z}\) commute with \(\partial _{s_{h}}\) for \(h \in [1, k-1]\) in \(\mathscr {D}_H\).
Remember that t is a fixed complex parameter.
References
Barlet, D.: On partial differential operators annihilating trace functions (to appear)
Barlet, D.: Développement asymptotique des fonctions obtenues par intégration sur les fibres. Invent. Math. 68(1), 129–174 (1982)
Barlet, D.: Fonctions de type trace. Ann. Inst. Fourier 33(2), 43–76 (1983)
Barlet, D., Maire, H.-M.: Asymptotique des intégrales-fibres. Ann. Inst. Fourier 43(5), 1267–1299 (1993)
D. Barlet and J. Magnússon Cycles analytiques complexes I: théorèmes de préparation des cycles, Cours Spécialisés 22, Société Mathématique de France (2014)
Bjork, J-E .: Analytical \({\cal{D}}\)-Modules and Applications, Mathematics and Its Applications. Kluwer Academic Publishers, vol. 247 (1993)
Kashiwara, M.: On the holonomic systems of linear differential equations II. Invent. Math. 49, 121–135 (1978)
Kashiwara, M.: \(\cal{D}\)-modules and microlocal calculus, Translations of Mathematical Monographs 217. American Math, Soc (2003)
Kashiwara, M.: The Riemann–Hilbert problem for holonomic systems. Publ. RIMS, Kyoto University 20, 319–365 (1984)
Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der Math. Wiss. 292, Springer (1990)
Kashiwara, M., Schapira, P.: Moderate and formal cohomology associated with constructible sheaves, Mémoires de la SMF 64, Société Mathématique de France (1996)
Loeser, F.: Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants. Ann. Sci. École Norm. Sup. 22(3), 435–471 (1989)
Sabbah, C.: Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques. Compos. Math. 64(2), 213–241 (1987)
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The research of T. Monteiro Fernandes was supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2019.
