Celebratio Mathematica — Fuchs — Complete Bibliography

Complete Bibliography

Works connected to Ernest Borisovich Vinberg

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D. Fuchs and A. Schwarz: “Matrix Vieta theorem,” pp. 15–22 in Lie groups and Lie algebras: E. B. Dynkin’s seminar. Edited by S. G. Gindikin and E. B. Vinberg. American Mathematical Society Translations. Series 2 169. American Mathematical Society (Providence, RI), 1995. This book is also no. 26 in the Advances in Mathematics series. MR 1364450 Zbl 0837.15011 ArXiv math/9410207 incollection

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-\( k \) matrices.

Specifically, we prove that if \( X_1,\dots \), \( X_n \) are solutions of an algebraic matrix equation \[ X^n + A_1 X^{n-1} + \cdots + A_n = 0 ,\] independent in the sense that they determine the coefficients \( A_1,\dots \), \( A_n \), then the trace of \( A_1 \) is the sum of the traces of the \( X_i \), and the determinant of \( A_n \) is, up to a sign, the product of the determinants of the \( X_i \). We generalize this to arbitrary rings with appropriate structures.

This result is related to and motivated by some constructions in non-commutative geometry.

Evgeniĭ Borisovich Dynkin	Related
Simon Grigorevich Gindikin	Related
Albert S. Schwarz	Related	Volume
Ernest Borisovich Vinberg	Related

B. L. Feigin and D. B. Fuchs: “Cohomologies of Lie groups and Lie algebras,” pp. 125–223 in Lie groups and Lie algebras, II. Edited by E. B. Vinberg. Encyclopaedia of Mathematical Sciences 21. Springer (Berlin), 2000. Translation of Russian original published in Seriya sovremennye problemy matematiki 21 (1988). MR 1756408 Zbl 0931.17014 incollection

Boris L. Feigin	Related
Ernest Borisovich Vinberg	Related