Rolle Models in the Real and Complex World

Dmitry Novikov () and Sergei Yakovenko ()
Additional contact information
Dmitry Novikov: The Weizmann Institute of Science, Department of Mathematics
Sergei Yakovenko: The Weizmann Institute of Science, Department of Mathematics

Chapter Chapter 6 in Handbook of Geometry and Topology of Singularities V: Foliations, 2024, pp 281-334 from Springer

Abstract: Abstract Numerous problems of analysis (real and complex) and geometry (analytic, algebraic, Diophantine e.a.) can be reduced to calculation of the “number of solutions” of systems of equations, defined by algebraic equalities and differential equations with algebraic right hand sides (both ordinary and Pfaffian). In the purely algebraic context the paradigm is given by the Bézout theoremBézout theorem: the number of isolated solutions of a system of polynomial equations of degree ≤ d $$\le d$$ in the n-dimensional space does not exceed d n $$d^n$$ , the bound polynomial in d and exponential in n. This bound is optimal: if we count solutions properly (i.e., with multiplicities, including complex solutions and solutions on the infinite hyperplane), then the equality holds. This paradigm can be generalized for the transcendental case as described above. It turns out some counting problems admit similar bounds depending only on the degrees and dimensions, whereas other can be treated only locally, i.e., admit bounds for the number of solutions in some domains of limited size. There is only one class of counting problems, which admits global bounds, but besides the degree and dimension, the answer depends on the heightHeight of the corresponding Pfaffian system. The unifying feature for these results is the core fact that lies at the heart of their proofs. This fact can be regarded as a variety of distant generalizations of the Rolle theorem known from the undergraduate calculus, claiming that between any two roots of a univariate differentiable function on a segment must lie a root of its derivative. We discuss generalizations of the Rolle theorem for vector-valued and complex analytic functions (none of them straightforward) and for germs of holomorphic maps.

Date: 2024
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-52481-3_6

Ordering information: This item can be ordered from
http://www.springer.com/9783031524813

DOI: 10.1007/978-3-031-52481-3_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().