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 .00347902) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011666) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00568224) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00966026) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0146838) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00692882) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00544266) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00545372) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0009614) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00068846) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00071674) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00467406) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0052403) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0069989) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00715112) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00465166) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00639378) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00536072) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0058131) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00613882) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003318) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009256) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002328) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000328) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009168) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002594) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0033863) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008894) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007108) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00059282) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0005251) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00208174) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00243002) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00040576) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00033446) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0006736) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00064318) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00272334) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00302742) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000306) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003508) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004324) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004424) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0162383 #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 .00353718) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011654) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00573724) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0686746) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0149857) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00696914) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00549086) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00552242) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00095168) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00075364) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00069672) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00480796) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00537324) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0071163) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00730836) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00473056) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0066518) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00543632) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00592866) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00626798) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003044) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009352) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000267) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003604) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009654) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002422) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00340852) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009864) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007376) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00058774) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00053112) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0020993) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0024387) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00041298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00033524) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00069682) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00065644) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00274154) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00312458) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003342) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003564) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0137636) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0126991) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00058818) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00056306) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014452) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014016) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003212) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003766) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0168757 #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.