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 (10[1])of g’s (“gees”) such that the leading terms of g and I generate the same ideal; (10[1])those instances of these sets are computed using Buchberger’s (10[1])algorithm. (10[2]-5[1])These sets (10[1])name a theorem (10[1])that states the polynomial ring of a Noetherian (10[2])(“no-THEER-ee-in”) ring (10[1])is Noetherian, (10[2]-5[1])as (-5[2])proven by Hilbert. (10[3])The Legendre (10[2])polynomials (10[2])form a complete (10[1])one of these (10[1])sets (10[2])for (10[1])L2 (10[2]-5[1])(“L-two”) functions on the open (10[2])unit (-5[1])interval. (-5[1])One of these (-5[1])sets is found (10[2])by setting [read (10[2])slowly] u-sub-one (10[1])to k-sub-one, then (10[1])iteratively setting u-sub-n (10[1])to (10[1])k-sub-n minus a sum (10[1])of projections (-5[1])of k-sub-n (10[4]-5[2])onto u-sub-i. (10[2])An orthonormal one (10[2])of (10[2])these sets (10[2])is (10[1])constructed (10[1])using the (10[2]-5[1])Gram–Schmidt process. For 10 points, (10[1])name these (-5[1])linearly (10[1])independent sets (10[2])that span (10[1])a (10[2])vector (10[8]0[17])space. ■END■