From Hilbert's 13th problem to essential dimension and back
Summary: This paper has two goals. The first is to survey some of the research on essential dimension in Sections 2--7. This survey is not comprehensive; it is only intended to convey the flavor of the subject and sample some of its highlights. My second goal for this paper is to define the notion of resolvent degree of an algebraic group in Section 8, building on the work of Farb and Wolfson [\textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104); \textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104)] but focusing on connected, rather than finite groups. The quantity \(\mathrm{rd}(n)\) defined above is recovered in this setting as \(\mathrm{rd}(S_n)\). For more comprehensive surveys of essential dimension and resolvent degree, see [\textit{A. S. Merkurjev}, Transform. Groups 18, No. 2, 415--481 (2013; Zbl 1278.14066); \textit{Z. Reichstein}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 162--188 (2011; Zbl 1232.14030)] and [\textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104)], respectively.
scientific article; zbMATH DE number 7475657
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