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            <title xml:lang="en">Symmetry in Multivariate Ideal Interpolation</title>
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                <term xml:lang="en">Interpolation</term>
                <term xml:lang="en">Symmetry</term>
                <term xml:lang="en">H-basis</term>
                <term xml:lang="en">Vandermonde matrix</term>
                <term xml:lang="en">Macaulay matrix</term>
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              <p>An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes,while  Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that  the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is  exploited computationally  at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly  to compute fundamental invariants and equivariants of  a reflection group.</p>
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