Invariant differential polynomials
https://doi.org/10.22405/2226-8383-2023-24-4-212-238
Abstract
Based on the method proposed in the article for solving the so-called (𝑟, 𝑠)-systems of linear equations proven that the orders of homogeneous invariant differential operators 𝑛 of smooth real functions of one variable take values from 𝑛 to (𝑛(𝑛+1))/2 , and the dimension of the space of all such operators does not exceed 𝑛!. A classification of invariant differential operators of order 𝑛 + 𝑠 is obtained for 𝑠 = 1, 2, 3, 4, and for 𝑛 = 4 for all orders from 4 to 10. The only, up to factors, homogeneous invariant differential operators of the smallest order 𝑛 and the largest order (𝑛(𝑛+1))/2 are given, respectively, by the product of the 𝑛 first differentials (𝑠 = 0 ) and
the Wronskian (𝑠 = (𝑛 − 1)𝑛/2). The existence of nonzero homogeneous invariant differential operators of order 𝑛 + 𝑠 for 𝑠 <((1+√5)/2)*(𝑛 − 1) is proved.
About the Author
Fyodor Mikhailovich Malyshev
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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Review
For citations:
Malyshev F.M. Invariant differential polynomials. Chebyshevskii Sbornik. 2023;24(4):212-238. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-212-238
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