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Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert

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Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert To cite this version: Erick Rodriguez Bazan, Evelyne Hubert. Ideal Interpolation, H-Bases and Symmetry. ISSAC 2020 - International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, Greece. 10.1145/3373207.3404057. hal-02482098v2 HAL Id: hal-02482098 https://hal.inria.fr/hal-02482098v2 Submitted on 7 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected] ABSTRACT for each instantiated interpolation problem, that is both invariant Multivariate Lagrange and Hermite interpolation are examples of and of minimal degree. An interpolation space for Λ identifies with ideal interpolation. More generally an ideal interpolation problem the quotient space K»x¼/I. Hence a number of operations related is defined by a set of linear forms, on the polynomial ring, whose to I can already be performed with a basis of an interpolation kernels intersect into an ideal. space for Λ: decide of membership to I, determine normal forms For an ideal interpolation problem with symmetry, we address of polynomials modulo I and compute matrices of multiplication the simultaneous computation of a symmetry adapted basis of the maps in K»x¼/I. Yet it has also proved relevant to compute Gröbner least interpolation space and the symmetry adapted H-basis of bases or H-bases of I. the ideal. Beside its manifest presence in the output, symmetry is Initiated in [26], for a set Λ of point evaluations, computing a exploited computationally at all stages of the algorithm. Gröbner basis of I found applications in the design of experiments [29, 30]. As pointed out in [25], one can furthermore interpret the CCS CONCEPTS FGLM algorithm [10] as an instance of this problem. The linear forms are the coefficients, in the normal forms, of the reduced • Computing methodologies → Symbolic and algebraic algo- monomials. The alternative approach in [11] can be understood rithms; similarly. KEYWORDS The resulting algorithm then pertains to the Berlekamp-Massey- Interpolation; Symmetry; Representation Theory; Group Action; Sakata algorithm and is related the multivariate version of Prony’s H-basis; Macaulay matrix; Vandermonde matrix problem to compute Gröbner bases, border bases, or H-bases [1, 28, 35, 36] 1 INTRODUCTION All ,the above mentioned algorithms and complexity analyses Preserving and exploiting symmetry in algebraic computations is a heavily depend on a term order and basis of monomials. These challenge that has been addressed within a few topics and, mostly, are notoriously not suited for preserving symmetry. Our ambition for specific groups of symmetry; For instance interpolation and in this paper is to showcase how symmetry can be embedded in symmetric group [23], cubature [4, 14], global optimisation [17, 32], the representation of both the interpolation space and the repre- equivariant dynamical systems [15, 20] and solving systems of sentation of the ideal. This is a marker for the more canonical polynomial equations [12, 13, 16, 19, 21, 31, 38]. In [33] we addressed representations. multivariate interpolation and in this article we go further with ideal The least interpolation space, defined in6 [ ], and revisited in [33] interpolation. We provide an algorithm to compute simultaneously is a canonically defined interpolation space. It serves here as the a symmetry adapted basis of the least interpolation space and a canonical representation of the quotient of the polynomial algebra symmetry adapted H-basis of the associated ideal. In addition to by the ideal. It has great properties, even beyond symmetry, that being manifest in the output, symmetry is exploited all along the cannot be achieved by a space spanned by monomials. In [33] algorithm to reduce the size of the matrices involved, and avoid we freed the computation of the least interpolation space from sizable redundancies. Based on QR-decomposition (as opposed to its reliance on the monomial basis by introducing dual bases. We LU-decomposition previously) the algorithm also lends itself to pursue this approach here for the representation of the ideal by H- numerical computations. bases [24, 27]. Where Gröbner bases single out leading terms with a Multivariate Lagrange, and Hermite, interpolation are examples term order, H-bases work with leading forms and the orthogonality of the encompassing notion of ideal interpolation, introduced in [2]. with respect to the apolar product. The least interpolation space They are defined by linear forms consisting of evaluation atsome then reveals itself as the orthogonal complement of the ideal of nodes, and possibly composed with differential operators, without leading forms. gaps. More generally a space of linear forms Λ on the polynomial As a result, computing a H-basis of the interpolation ideal is achieved with linear algebra in subspaces of homogeneous polyno- ring K»x¼ = K»x1;:::; xn¼ is an ideal interpolation scheme if mials of growing degrees. Yet we shall first redefine the concepts at Ù I = ker λ = fp 2 K»x¼ : λ¹pº = 0; for all λ in Λg (1) play in an intrinsic manner, contrary to the computation centered λ 2Λ approach in [27, 34]. The precise algorithm we shall offer to com- is an ideal in K»x¼. In the case of Lagrange interpolation, I is the pute H-bases somehow fits in the loose sketch proposed in[5]. Yet ideal of the nodes and is thus a radical ideal. we are now in a position to incorporate symmetry in a natural way, If Λ is invariant under the action of a group G, then so is I. In refining the algorithm to exploit it; A totally original contribution. [33] we addressed the computation of an interpolation space for Λ Symmetry is preserved and exploited thanks to the block diago- i.e., a subspace of the polynomial ring that has a unique interpolant nal structure of the matrices at play in the algorithms. This block diagonalisation, with predicted repetitions in the blocks, happens is invertible. This latter is to be interpreted as the matrix in the when the underlying maps are discovered to be equivariant and bases P and the dual of L of the restriction of the Vandermonde expressed in the related symmetry adapted bases. The case of the operator w : K»x¼ ! Λ∗ such that w¹pº¹λº = λ¹pº. This is the adjoint Vandermonde matrix was settled in [33]. In this paper, we also need of embedding Λ ,! K»x¼∗ and hence is surjective. the matrix of the prolongation map, knowned in the monomial All along this paper we shall assume that basis as the Macaulay matrix. Figuring out the equivariance of this I = kerw = \λ 2Λ ker λ map is one of the original key results of this paper. Λ = he ;:::; e i I The paper is organized as follows. In Section 2 we define ideal is an ideal. When for instance ξ1 ξr K then is the ideal of the points fξ1;:::; ξr g ⊂ K»x¼. One sees in general that interpolation and explain the identification of an interpolation ∗ space with the quotient algebra. In Section 3 we review H-bases and dim K»x¼/I = dim Λ = dim Λ =: r: discuss how they can be computed in the ideal interpolation setting. With Q = fq1;:::; qr g ⊂ K»x¼, we can identify K»x¼/I with In Section 4 we provide an algorithm to compute simultaneously a hQiK if hQiK ⊕ I = K»x¼. With a slight shortcut, we say that Q is basis of the least interpolation space and an orthogonal H-basis of a basis for K»x¼/I. the ideal. In Section 5 we show how the Macaulay matrix can be Proposition 2.1. Q = fq1;:::; qr g ⊂ K»x¼ spans an interpola- block diagonalized in the presence of symmetry. This is then applied tion space for Λ iff it is a basis for the quotient K»x¼/I: in Section 6 to obtain an algorithm to compute simultaneously a symmetry adapted basis of the least interpolation space and a Proof. If Q = fq1;:::; qr g is a basis of K»x¼/I then for any symmetry adapted H-basis of the ideal. All along the paper, the p 2 K»x¼ there is a q 2 hq1;:::; qr iK such that p ≡ q mod I. Hence definitions and notations comply with those in [33]. λ¹pº = λ¹qº for any λ 2 Λ and thus hQiK is an interpolation space for Λ. Conversely if hq1;:::; qr iK is an interpolation space for Λ 2 IDEAL INTERPOLATION then fq1;:::; qr g are linearly independent modulo I and therefore a basis for K»x¼/I. Indeed if q = a1q1 + ::: + ar qr 2 I then any In this section, we consider the ideal interpolation problem and interpolation problem has multiple solutions in hQiK, i.e, if p is explain the identification of an interpolation space with the quotient the solution of ¹Λ; ϕº so is p + q, contradicting the interpolation algebra. We recall that the least interpolation space is the orthogonal uniqueness on hQiK. □ complement of the ideal of the leading forms, I0. K denotes either C or R.
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