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. (-5[1])These sets name a theorem that states the polynomial ring of a Noetherian (“no-THEER-ee-in”) ring is Noetherian, as proven by Hilbert. The Legendre polynomials form a complete (10[1])one of these sets (10[2])for L2 (“L-two”) functions on the open unit (-5[1])interval. One of these sets is found by setting [read slowly] u-sub-one (10[1])to k-sub-one, then iteratively setting u-sub-n to k-sub-n (10[1])minus a sum of projections of k-sub-n onto u-sub-i. An orthonormal (10[1])one of these sets is constructed using the Gram–Schmidt process. For 10 points, name these linearly independent sets that span a vector (10[1])space. ■END■ (10[1])

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