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 .0047814) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00013454) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00759104) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0130243) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0200128) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0092561) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00729532) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00769304) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00118298) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00091418) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00115692) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00652078) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00697176) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00947316) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00972038) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00636946) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0085451) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00729578) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00788042) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00856414) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002958) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000932) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004382) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000323) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011984) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002416) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00441986) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011342) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007188) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00088218) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00076638) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00286872) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00306572) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00051454) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00036656) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00083174) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00091594) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00369002) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00406904) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004298) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004636) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00003758) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004204) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0184588 #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 .0043943) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00014124) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0075853) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0124258) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0205327) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00896254) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0078055) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0072401) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00117756) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00085724) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00084036) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00635384) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00703218) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00936942) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00894126) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0067019) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0087098) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00737256) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00739732) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0077731) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000298) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00023756) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000424) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004586) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000907) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002256) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00454332) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011468) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008588) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0007391) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00094336) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0026614) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00327394) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00052974) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00035882) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00084644) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00082466) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00371808) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0040139) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002666) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000305) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0170456) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0156581) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00086726) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00077226) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00019936) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00015108) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003242) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003996) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0169346 #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.