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 .0010242) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028752) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00160866) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0026959) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00419852) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00186994) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00149712) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00154885) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00029821) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000209784) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000202083) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00141342) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00159662) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00204398) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00212816) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00136154) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00186429) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156268) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173635) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00182853) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008265) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021733) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004736) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005232) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019437) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005933) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000899825) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022556) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017224) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000177596) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000165673) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000594052) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000709676) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000113298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000092639) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000203657) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000193653) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000798504) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000906214) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006068) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006771) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000009948) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00000886) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00443617 #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 .00108124) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000031265) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00171381) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00285632) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00441519) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00199896) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00155402) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0015991) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000304311) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000211839) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000216008) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00137007) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0015993) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00208355) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00214464) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00133621) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00179036) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00147365) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00163841) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00176407) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005545) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018163) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005742) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005124) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001811) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004913) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000845087) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019424) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016918) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000167917) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000160319) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000569484) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000660407) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00010894) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000084915) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000204706) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000192705) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00074304) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000844525) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005188) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005196) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00352432) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00324966) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000140208) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000134893) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000038005) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000035726) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005575) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005808) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00419386 #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.