Round 1: Tossup 17

One of these sets is most efficiently computed for a set in the graded reverse lexicographic monomial ordering. 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 that states the polynomial ring of a Noetherian (10[1])(“no-THEER-ee-in”) ring is Noetherian, (10[1])as proven by Hilbert. The Legendre polynomials (10[1])form a complete one of these sets for L2 (10[2]-5[1])(“L-two”) functions on the open unit interval. (-5[1])One of these sets is found by setting [read slowly] u-sub-one to k-sub-one, (10[1])then iteratively setting 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 is constructed using (10[1])the Gram–Schmidt process. For 10 points, name these (10[1])linearly independent sets that span a vector space. (10[2])■END■ (0[2])

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