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 .00113907) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034693) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00208639) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00348613) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00549057) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00247172) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00195033) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0020057) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000376843) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000277354) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00024306) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0015933) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0018532) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00243706) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00252183) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00216854) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00179359) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00198215) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00209928) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008112) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033303) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006912) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009986) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028232) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006887) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00113207) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033404) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002543) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000222463) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000212494) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000761802) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000866061) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00016092) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000127755) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00022747) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00023364) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000934375) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0010531) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012615) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000101) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000020154) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000013886) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00575039 #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 .00113188) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000046868) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0020748) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00348223) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00552691) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00243645) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00195796) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00200959) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000363047) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00028999) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000248156) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00158578) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00180857) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00242345) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00250957) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00157647) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00217565) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00175388) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0019675) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00213659) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013174) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028059) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007073) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009956) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000044497) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007715) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00113913) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028001) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024171) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000240191) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021571) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000762071) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000870674) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00014773) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000126218) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000240171) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000223278) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000917781) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00104194) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008077) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001056) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00420351) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00374701) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000178882) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00018353) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000046257) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000038602) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012601) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011487) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00489889 #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.