Decompositions of the Kuchment-Lvin Polynomials
Douglas Weathers · DigitalCommons (California Polytechnic State University) · 2015
While studying the Radon transform in the context of medical tomography, Peter Kuchment and Sergey Lvin encountered an infinite family of differential polynomials fn,λ(u) where n is a positive integer, A is a complex parameter, and u is a smooth function with derivatives u'u",..., u(m) etc. It was discovered that (i) if u' = λu, then fn,λ(u) = 0 for all n and that (ii) if u" = λ2u, then fn,λ(u) = 0 for all odd n. This leads to a natural question (iii): if m ≥ 3, does u(m) = λmu imply that fn,λ(u) = 0 for a progression of n whose common difference is ml We divide fn,λ(u) into a linear part, quadratic part, etc. according the degree of the terms. With an eye to the algebraic structure of the fn,λ(u), we then use combinatorics to prove that all parts of the polynomial vanish if u' = λu and provide an alternate proof of identity (i). We then apply the same techniques to answer question (iii) negatively for nonzero A and the linear part of the polynomial: If m ≥ 3 is an integer such that u(m) = λmu and the linear part of fn,λ(u) vanishes for an integer n > 2, then it must be the case that u' = λu or u" = λ2u.