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 .0037532) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00012804) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00595068) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .010124) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0155452) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0764245) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00607804) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00595556) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00103454) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00075054) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007834) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0050829) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00573356) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0074222) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0077545) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00510982) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00685838) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0058753) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0063278) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00678544) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003648) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010168) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000276) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003386) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001058) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002924) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00368568) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010228) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008606) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00061856) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00055568) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00225812) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00259214) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00043324) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00034966) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0007397) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00070848) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00294814) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0032151) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003178) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003534) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000434) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000478) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .017493 #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 .00390254) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00012976) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00605028) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0105546) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0159167) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00752598) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00591658) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00587246) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0010464) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007742) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007798) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00514142) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00576476) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0074747) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00770002) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00495322) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00674678) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00561526) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00623194) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0065307) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003258) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010126) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003406) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003402) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009948) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003362) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00349634) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001028) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009172) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00062274) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0005584) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00216574) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00259658) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00042822) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0003476) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00071974) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00069174) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00279686) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00319976) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002848) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003644) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0137483) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0126497) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0005777) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00057244) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00013986) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0001348) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003966) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003916) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0167939 #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.