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 .00135556) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039494) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00238293) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00371605) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00590056) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00249587) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00199634) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00214913) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000430022) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000279194) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000277296) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00167719) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00201495) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00265599) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00274959) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0017162) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00233013) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00195156) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00216512) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00228778) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000762) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024502) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008088) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006858) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025422) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000691) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117196) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025834) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024976) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00028766) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000249374) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00078288) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000922996) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000158014) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000122912) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000248082) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024209) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000979388) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00112483) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006772) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007816) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010966) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00001178) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00493713 #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 .00133668) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003792) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00238945) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00373768) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00591799) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00251408) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00202143) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00213787) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000430384) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000280764) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000282116) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .001724) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207712) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00266269) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00279688) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0130587) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00236456) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00196783) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00216127) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231941) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007186) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023536) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000676) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007076) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024082) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006768) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00118317) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025652) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024832) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000273766) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000248284) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000798152) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000918306) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000157814) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000122818) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000259264) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000243572) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0009775) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00111802) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006636) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007824) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00488366) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00437555) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000220544) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000217782) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00005243) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000052506) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008492) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000778) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00494965 #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.