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 .000884474) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027066) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00133711) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00274417) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00205692) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00116555) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00127285) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0002238) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000151261) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000150113) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00103929) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00116189) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0016517) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00136511) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00113482) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00121353) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00143796) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027906) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022325) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004963) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000721) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000723017) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032873) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023825) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000221942) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000597723) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000597165) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000088307) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000069162) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000151333) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000569943) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00081886) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007086) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000827) #primes = 7 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010646) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00315414 #minprimes=6 #computed=8 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 .000881482) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000030892) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00130172) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00316015) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00217826) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0014255) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00155575) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000229429) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000154772) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000152145) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00114842) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00112358) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00153328) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00149942) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00117071) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00132924) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00133008) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007737) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018783) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007149) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006609) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00075581) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019712) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016333) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000150247) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000531122) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000571788) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000092344) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000072319) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000151374) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000594127) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000674868) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005994) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006899) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .00279784) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .00012377) #primes = 7 #prunedViaCodim = 0 Strategy: Linear (time .000027779) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007192) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00274722 #minprimes=6 #computed=8 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.