University of Paderborn Department of Mathematics Diploma Thesis ...

4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 105d:= 1;deg:= degree(p);pList:= [p, 0 $ deg];/* ’pList’ will contain the set of the ’Delta polynomials’, i.e.pList = [ p,Delta^1(z) = p(z) - p(z-1),Delta^2(z) = Delta^1(z) - Delta^1(z-1). . .. . .. . .Delta^m(z) = Delta^(m-1)(z) - Delta^(m-1)(z-1)]*/while poly(expand(p(z) - p(z-1)), [z]) poly(0, [z]) dod:= d + 1;p:= poly(expand(p(z) - p(z-1)), [z]);pList[d]:= p;end_while;/* Compute the unique lexicographic ideal L associated to theHilbert polynomial ’pList[d]’ in R = K[x_0,...,x_{n-d+1}]. */L:= compute_L_p(pList[d], n-d+1);/* Use procedure ’compute_ideals’ to compute all saturated stableideals with the same Hilbert polynomial as L in R.From now on we collect all computed ideals in the set M. */M:= compute_ideals(L, n-d+1);d:= d-1;/* Treat the special cases for constant and linear Hilbert polynomials:deg = 0 -- the Hilbert polynomial is constant. Hence, the doublesaturation of all ideals computed equals the wholepolynomial ring.deg = 1 -- the Hilbert polynomial is linear. Thus, the set Mcontains all possible double saturations of the idealsassociated to the Hilbert polynomial at this time.The case with deg > 1 is treated below within the for-loop */if deg = 0 and args(0) = 3 and args(3) = hold(All) thenprint(Unquoted, "constant Hilbert polynomial: the double saturation");