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 .00349142) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010384) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00553816) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00929242) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0141844) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00655988) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0051925) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0052438) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00095842) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00071174) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007022) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00446136) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00523876) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00675052) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00693644) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0045348) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00618724) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00515312) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00568638) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00601484) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002712) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000787) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002176) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002504) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007388) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002434) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0031798) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000744) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00006036) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0005598) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00051848) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00198996) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00232362) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00040764) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00029058) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00067478) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00068086) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00260674) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00296896) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002194) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002254) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00003212) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000308) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0134364 #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 .00351392) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010312) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00549046) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0093523) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0141577) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00661798) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00523864) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00524906) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00094522) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00070386) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00067446) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00456902) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00518176) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0570256) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00728128) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00465196) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0062468) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00521436) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00581188) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0061341) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002712) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009568) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002188) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007814) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002516) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00323502) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007324) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00006394) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00057312) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0005198) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0020157) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00232904) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00039994) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00032878) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0006561) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00063504) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0026268) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00292594) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002142) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000245) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0130827) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0119716) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00061886) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00060626) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0001332) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014344) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002652) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002516) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0137782 #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.