Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either QQ or ZZ/p, return a list of minimal primes of the ideal.
i1 : R = ZZ/32003[a..e] o1 = R o1 : PolynomialRing |
i2 : I = ideal"a2b-c3,abd-c2e,ade-ce2" 2 3 2 2 o2 = ideal (a b - c , a*b*d - c e, a*d*e - c*e ) o2 : Ideal of R |
i3 : C = minprimes I; |
i4 : netList C +---------------------------+ o4 = |ideal (c, a) | +---------------------------+ | 2 3 | |ideal (e, d, a b - c ) | +---------------------------+ |ideal (e, c, b) | +---------------------------+ |ideal (d, c, b) | +---------------------------+ |ideal (d - e, b - c, a - c)| +---------------------------+ |ideal (d + e, b - c, a + c)| +---------------------------+ |
i5 : C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2) Strategy: Linear (time .00196867) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00005977) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00337519) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00528135) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0081821) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00359291) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00281373) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00294609) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00056747) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000377138) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000395632) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00243711) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .002881) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00380236) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00388092) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00248453) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00339873) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0027997) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00308557) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0033252) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001617) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035374) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009492) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009576) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033164) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009526) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00173034) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033928) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032898) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000370734) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000326514) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00109029) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00128306) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000211272) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000162602) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000358208) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00035014) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00141094) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00160105) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010144) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009534) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000017884) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000015276) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00718151 #minprimes=6 #computed=10 2 3 o5 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b), ------------------------------------------------------------------------ ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)} o5 : List |
i6 : C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2) Strategy: Linear (time .00194264) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000057896) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00336378) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0053316) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00825004) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00359493) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0028788) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0029825) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00056973) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00037947) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00038005) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00248293) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00294903) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00379752) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00396887) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .021242) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00343957) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00285261) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00311717) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00331236) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012794) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037544) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001094) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000962) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033494) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000951) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00173282) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035744) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034194) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000367378) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000323516) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00109157) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00132413) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000212546) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000160846) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000374996) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00035043) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00141392) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00159517) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001044) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001084) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00718587) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00642638) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000317192) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000316358) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000083562) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000077732) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001445) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010694) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00723709 #minprimes=6 #computed=10 2 3 o6 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b), ------------------------------------------------------------------------ ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)} o6 : List |
This will eventually be made to work over GF(q), and over other fields too.