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 .00151196) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004981) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00256228) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00431685) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00667584) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00301758) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0023873) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00244059) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000458674) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000322695) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000302576) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00200741) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00232524) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00306548) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00315293) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00202123) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00274157) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227451) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00250187) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0027006) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009153) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034102) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008037) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008742) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028251) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013192) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00136039) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028805) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034946) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000263639) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000258723) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000910672) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00106209) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000185541) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000147965) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000286161) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000267019) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00118089) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00132026) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008711) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008979) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000019541) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000013327) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00581203 #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 .00151303) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000048755) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00256271) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00429515) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00669776) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00302547) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00239873) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00242844) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000466988) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00031607) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000308141) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00200323) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00232144) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00306999) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00315968) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0020141) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00275685) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00229131) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00253063) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00266452) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013555) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033328) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008076) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010446) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002833) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008543) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00135627) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029061) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026086) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000263264) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000255138) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000917863) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00106315) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000181663) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000149235) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000283084) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000279439) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0011677) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00133353) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008435) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000916) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00549334) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00517055) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000276998) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000255225) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000060439) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000055023) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000921) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009837) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00580812 #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.