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 .0013899) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000051292) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0025327) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00440367) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00637425) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00277566) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00234801) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00242029) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000470412) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000310194) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000273581) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00188373) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0022268) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00289449) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00303762) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00190769) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00377254) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215023) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00229977) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00252808) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010188) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000048199) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008423) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010722) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033152) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008578) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0014621) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037539) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029598) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000251741) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000230215) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000924009) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00102199) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00020856) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000180928) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00030035) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000259245) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00109554) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00131104) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014034) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010778) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000020188) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000015243) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00645597 #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 .00131595) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000044711) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00237669) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00403012) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00665679) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00294098) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00268403) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00246302) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00044486) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000321386) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000297291) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227924) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00252624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00286097) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00293432) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0017853) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00268266) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215941) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00239249) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00262014) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014324) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000043606) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007751) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010559) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038673) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007859) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00143275) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003258) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027788) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000291163) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000273676) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000879798) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00106706) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000184187) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000147834) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000280357) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000282605) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00109926) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00122402) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009816) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011479) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00483444) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00432122) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000205294) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000192227) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000044758) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000043847) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010082) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013196) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00593404 #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.