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Register a new userA new proof of the Gasca-Maeztu conjecture for n = 5
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Added by
Gagik Vardanyan
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
G. K. Vardanyan
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An $n$-correct node set $mathcal{X}$ is called $GC_n$ set if the fundamental polynomial of each node is a product of $n$ linear factors. In 1982 Gasca and Maeztu conjectured that for every $GC_n$ set there is a line passing through $n+1$ of its nodes.So far, this conjecture has been confirmed only for $nle 5.$ The case $n = 4,$ was first proved by J. R. Bush in 1990. Several other proofs have been published since then. For the case $n=5$ there is only one proof: by H. Hakopian, K. Jetter and G. Zimmermann (Numer Math $127,685-713, 2014$). Here we present a second, much shorter and easier proof.
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Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {mathcal L}^{-1} {mathcal F}(hat{u}) + {mathcal L}^{-1} {mathcal G}(w)$, where ${mathcal L}$ is a linearized operator, ${mathcal F}(hat{u})$ is a residual, and ${mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $| {mathcal L}^{-1} {mathcal F}(hat{u}) |$ and $| {mathcal L}^{-1}{mathcal G}(w) |$ play major roles in the verification procedures. In this paper, using a similar concept as the `Schur complement for matrix problems, we represent the inverse operator ${mathcal L}^{-1}$ as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as ${mathcal L}^{-1}$ are presented in the appendix.
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