The famous problem enunciated by the illustrious French mathematician Henri Poincaré (1854-1912) – the Center-Focus Problem, on which the great mathematicians of the world pondered for more than a century, was solved in Chișinău – an algebraic solution to which, we say it with immeasurable honor, contributed the formidable univ. prof., Dr. in Physics-Mathematical Sciences, Mihail POPA, and the young professor of TUM, Doctor in Mathematical Sciences, Victor PRICOP. Coincidentally or not, around Science Day, copies of the monograph “The Center and Focus Problem. Algebraic Solutions and Hypotheses”, signed by the two Moldovan mathematicians, who were the first in the world to demonstrate the solution of Poincaré’s Problem, arrived from the USA.
Thousands of works have been written on Poincaré’s Center-Focus Problem in various scientific centers around the world – France, Russia, Belarus, China, Great Britain, Canada, USA, their number approaching 100 in the Republic of Moldova only. A solution was offered by the university professor Mihail POPA, founder of the scientific school Lie Algebras and Differential Systems, who achieved an exceptional result and discovered an answer to a problem formulated almost 140 years ago.
The professor started by establishing the connection between the Lie algebras of Sophus Lie (1842-1899), Norway, and the graded algebras of invariants and comitants of acad. Constantin Sibirschi (1928-1990). His conclusions were based on the Generalized Center-Focus Problem for Differential Systems, while avoiding the calculation of focal quantities for each system, outlined by the Russian mathematician Alexander Lyapunov (1857-1918). The calculation included the application of the methods of Lie algebras and graded algebras of invariants and comitants, as well as the generating of functions and Hilbert series. As a result, a finite numerical estimate was obtained for the algebraically independent Lyapunov quantities, which participate in the solving of the Center-Focus Problem for any system of polynomial differential equations. Thus, for the first time, a reasoned hypothesis was formulated, according to which these found numbers form the finite upper bound of the number of Lyapunov quantities that completely solves the Generalized Center and the Focus Problem for each Polynomial Differential Systems.
Professor Mihail Popa was encouraged by prominent specialists in the field to continue his research in finalizing and explaining the solution of the Problem of the Center and the Focus. In these investigations, he was joined by his pupil Victor PRICOP, then a Doctoral student, currently a professor in the Department of Mathematics at the Technical University of Moldova, his scientific interests including Lie algebras and commutative graded algebras, generating functions and Hilbert series, and applications of algebras to polynomial differential systems. Together, the two researchers improved the initial hypothesis in the monograph “The Problem of the Center and the Focus: Algebraic Solutions and Hypotheses”, published in Russian – “Проблема центра и фокуса. Алгебраические решения и гипотезы”. Thus, the solution of the Generalized Center and the Focus Problem required more than 20 years of work, while simultaneously working on the research of other topics. The final solution required another 8 years of investigations.
The completed work of prof. Mihail POPA and Dr. Victor PRICOP was translated into English and presented at several publishing houses abroad. The best conditions were proposed by the Taylor & Francis Group Publishing House based in Great Britain, with a history of over 200 years, specialized in publishing scientific literature and journals, with 8 offices around the world, including 3 in the USA. The monograph “The Center and Focus Problem. Algebraic Solutions and Hypotheses”, assessed page by page and chapter by chapter, was recognized as an original scientific paper and signed for printing.
The cover presentation states: “The monograph focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincaré’s problem. Namely, there are given the upper estimations of the number of Poincaré-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.”
We express our respect and admiration for the authors! Congratulations!
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