Round 9: Tossup 17

One of these sets is most efficiently computed for a set in the graded reverse lexicographic monomial ordering. (10[1])One of these sets of an ideal I is defined as the set of g’s (“gees”) such that the leading terms of g and I generate the same ideal; those instances of these sets are computed using Buchberger’s algorithm. These sets name a theorem (10[1])that states the polynomial ring of a Noetherian (“no-THEER-ee-in”) ring is Noetherian, as (-5[1])proven by Hilbert. The Legendre (10[1])polynomials form a complete one of these (10[1])sets for L2 (“L-two”) functions on the open (10[1])unit interval. One of these sets is found (10[1])by setting [read slowly] u-sub-one to k-sub-one, then iteratively setting (10[1])u-sub-n to k-sub-n minus a sum of projections of k-sub-n onto u-sub-i. (10[1])An orthonormal one of these sets (10[1])is constructed using the Gram–Schmidt process. For 10 points, name these linearly independent sets that span a vector space. ■END■ (0[5])

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