Geodesic
Average number of solutions for systems of equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 2, pp. 35-47.

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For $n$ finite-dimensional spaces of smooth functions $V _i $ on a smooth $n$-dimensional manifold $X$, the systems of equations $ \{f_i = a_i \colon \: f_i \in V_i, \: a_i \in \mathbb{R}, \: i = 1, \ldots, n \} $ are considered. A connection is established between the average numbers of solutions and the mixed volumes of convex bodies. To do this, fixing Banach metrics of the spaces $ V_i $, we construct 1) measures in the spaces of systems of equations, and 2) Banach convex bodies in $X$, those. families of centrally symmetric convex bodies in the layers of the cotangent bundle $X$. It is proved that the average number of solutions is equal to the mixed symplectic volume of Banach convex bodies. The case of Euclidean metrics in the spaces $ V_i $ was previously considered. In this case, the Banach bodies are ellipsoid families.
Keywords: Banach space, Crofton formula, normal density, mixed volume. 
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B. Ya. Kazarnovskii. Average number of solutions for systems of equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 2, pp. 35-47. http://geodesic.mathdoc.fr/item/FAA_2020_54_2_a2/

[1] D. Akhiezer, B. Kazarnovskii, “Average number of zeros and mixed symplectic volume of Finsler sets”, Geom. Funct. Anal., 28:6 (2018), 1517–1547 | DOI | MR | Zbl

[2] D. Akhiezer, B. Kazarnovskii, The ring of normal densities, Gelfand transform and smooth BKK-type theorems, arXiv: 1907.00633

[3] S. Alesker, “Theory of valuations on manifolds: a survey”, Geom. Funct. Anal., 17:4 (2007), 1321–1341 | DOI | MR | Zbl

[4] S. Alesker, J. Bernstein, “Range characterization of the cosine tranform on higher Grassmanians”, Adv. Math., 184:2 (2004), 367–379 | DOI | MR | Zbl

[5] D. N. Bernshtein, “Chislo kornei sistemy uravnenii”, Funkts. analiz i ego pril., 9:3 (1975), 1–4

[6] D. Yu. Burago, S. Ivanov, “Izometricheskie vlozheniya finslerovykh mnogoobrazii”, Algebra i analiz, 5:1 (1993), 179–192 | MR | Zbl

[7] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969 | MR | Zbl

[8] J. Nash, “The imbedding problem for Riemannian manifolds”, Ann. of Math., 63:1 (1956), 20–63 | DOI | MR | Zbl

[9] R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, Encyclopedia of Math, and its Appl., no. 44, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[10] R. Schneider, “Crofton measures in projective Finsler Spaces”, Integral Geometry and Convexity (Wuhan, China, 18–23 October 2004), World Sci. Publ., Hackensack, NJ, 2006, 67–98 | DOI | MR | Zbl

[11] D. N. Zaporozhets, Z. Kabluchko, “Sluchainye opredeliteli, smeshannye ob'emy ellipsoidov i nuli gaussovskikh sluchainykh polei”, Zap. nauchn. sem. POMI, 408 (2012), 187–196