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-- test monomial ordering R = QQ[a..d] assert ( sort basis(0,3,R) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}} ) assert ( sort basis(0,3,R^2) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}} ) R = QQ[a..d,MonomialOrder => Position => Up] assert ( sort basis(0,3,R) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}} ) assert ( sort basis(0,3,R^2) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}} ) R = QQ[a..d,MonomialOrder => Position => Down] assert ( sort basis(0,3,R) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}}) assert ( sort basis(0,3,R^2) == matrix {{1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, d, c, b, a, d^2, c*d, b*d, a*d, c^2, b*c, a*c, b^2, a*b, a^2, d^3, c*d^2, b*d^2, a*d^2, c^2*d, b*c*d, a*c*d, b^2*d, a*b*d, a^2*d, c^3, b*c^2, a*c^2, b^2*c, a*b*c, a^2*c, b^3, a*b^2, a^2*b, a^3}} ) R = QQ[a..d,MonomialOrder => Lex] assert ( sort basis(0,3,R) == matrix {{1, d, d^2, d^3, c, c*d, c*d^2, c^2, c^2*d, c^3, b, b*d, b*d^2, b*c, b*c*d, b*c^2, b^2, b^2*d, b^2*c, b^3, a, a*d, a*d^2, a*c, a*c*d, a*c^2, a*b, a*b*d, a*b*c, a*b^2, a^2, a^2*d, a^2*c, a^2*b, a^3}}) assert ( sort basis(0,3,R^2) == matrix {{1, d, d^2, d^3, c, c*d, c*d^2, c^2, c^2*d, c^3, b, b*d, b*d^2, b*c, b*c*d, b*c^2, b^2, b^2*d, b^2*c, b^3, a, a*d, a*d^2, a*c, a*c*d, a*c^2, a*b, a*b*d, a*b*c, a*b^2, a^2, a^2*d, a^2*c, a^2*b, a^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, d, d^2, d^3, c, c*d, c*d^2, c^2, c^2*d, c^3, b, b*d, b*d^2, b*c, b*c*d, b*c^2, b^2, b^2*d, b^2*c, b^3, a, a*d, a*d^2, a*c, a*c*d, a*c^2, a*b, a*b*d, a*b*c, a*b^2, a^2, a^2*d, a^2*c, a^2*b, a^3}} ) R = QQ[a..d,MonomialOrder => RevLex,Global => false] assert ( sort basis(0,3,R) == matrix {{a^3, a^2*b, a^2*c, a^2*d, a^2, a*b^2, a*b*c, a*b*d, a*b, a*c^2, a*c*d, a*c, a*d^2, a*d, a, b^3, b^2*c, b^2*d, b^2, b*c^2, b*c*d, b*c, b*d^2, b*d, b, c^3, c^2*d, c^2, c*d^2, c*d, c, d^3, d^2, d, 1}}) assert ( sort basis(0,3,R^2) == matrix {{a^3, a^2*b, a^2*c, a^2*d, a^2, a*b^2, a*b*c, a*b*d, a*b, a*c^2, a*c*d, a*c, a*d^2, a*d, a, b^3, b^2*c, b^2*d, b^2, b*c^2, b*c*d, b*c, b*d^2, b*d, b, c^3, c^2*d, c^2, c*d^2, c*d, c, d^3, d^2, d, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a^3, a^2*b, a^2*c, a^2*d, a^2, a*b^2, a*b*c, a*b*d, a*b, a*c^2, a*c*d, a*c, a*d^2, a*d, a, b^3, b^2*c, b^2*d, b^2, b*c^2, b*c*d, b*c, b*d^2, b*d, b, c^3, c^2*d, c^2, c*d^2, c*d, c, d^3, d^2, d, 1}} ) R = ZZ assert( id_(R^2) == sort id_(R^2) ) R = QQ assert( id_(R^2) == sort id_(R^2) ) R = GF 9 assert( id_(R^2) == sort id_(R^2) ) R = ZZ[x] assert( id_(R^2) == sort id_(R^2) ) R = QQ[x] assert( id_(R^2) == sort id_(R^2) ) R = ZZ[] assert( id_(R^2) == sort id_(R^2) ) R = QQ[] assert( id_(R^2) == sort id_(R^2) ) R = degreesRing 0 assert( id_(R^2) == sort id_(R^2) )