Block-Symmetric Bases and Multidimensional Newton Formulas

• Datum: 2025
• 1. Verfasser: Ясельський, Андрій
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Zugang: http://mcm-math.kpnu.edu.ua/article/view/339155
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

In this work we develop a construction of block-symmetric invariants for sequences of multidimensional blocks and obtain multidimensional Newton formulas that link three natural systems of bases. The study relies on introducing block power sums as well as complete and elementary block-symmetric polynomials, for which generating functions are built and a fundamental vector identity is established. It is shown that the logarithmic form of this identity leads to a system of Newton-Girard recurrence relations with explicit combinatorial coefficients, which ensures a correct accounting of monomials in the multidimensional setting. The obtained relations are consistent with the classical formulas and do not require additional assumptions concerning the commutativity of transformations or any special normalization of coefficients. It is proved that transitions between the indicated bases have a triangular character with respect to the natural partial order on multi-indices, which entails the uniqueness of expansions and the invertibility of the corresponding linear transformations. The results include the statement that block power sums form a basis of the invariant subalgebra, while complete and elementary functions provide alternative expansions with clear coefficient conversion rules. Special attention is paid to the infinite-dimensional situation with truncation by the number of blocks. It is shown that the truncated representations form a sequence that is monotone with respect to inclusion and uniformly bounded on balls, which ensures uniform convergence on compact sets to the original invariant. On the basis of these properties, a conclusion is formulated about a minimal generating set for each fixed total degree. For any such degree, all invariants are generated by elements of the same degree from any of the three systems, while truncated series by the number of blocks converge to full polynomials uniformly on compact sets. The proposed scheme generalizes the classical theory of symmetric functions to the block case and forms a unified methodology for constructing bases, performing mutual transitions, and controlling convergence, which creates a foundation for further research in combinatorics and the algebraic analysis of invariants