Identification of an Algebraic Domain in Two Dimentions from a Finite Number of its Generalized Polorization Tensors

Habib Ammari (1) , Mihai Putinar (2) , Andries Steenkamp (1) , Faouzi Triki (3)

1 ETH-SAM - Seminar for Applied Mathematics
2 UC Santa Barbara - University of California [Santa Barbara]
3 EDP - Equations aux Dérivées Partielles

This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.

Domaines: Equations aux dérivées partielles [math.AP]

Habib Ammari, Mihai Putinar, Andries Steenkamp, Faouzi Triki. Identification of an Algebraic Domain in Two Dimentions from a Finite Number of its Generalized Polorization Tensors. Mathematische Annalen, 2019, 375, pp.1337-1354. ⟨10.1007/s00208-018-1780-y⟩. ⟨hal-01827232⟩