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 .00435882) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00013014) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00667398) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0115453) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0174382) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00817362) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00647346) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00646616) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00114052) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00083624) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00086628) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00564446) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00634382) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00843232) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00869098) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00563962) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00770218) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00646916) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00728966) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0077161) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004838) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001124) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004136) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007074) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009054) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002504) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00379492) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000971) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008204) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00066514) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00058018) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00222422) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00251854) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00042256) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00036498) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00074308) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00069866) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00286056) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00319924) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000264) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003136) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004424) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004352) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0164262 #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 .00369274) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00012268) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .006039) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0100871) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0156134) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00721854) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00566584) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00576538) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0010461) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00075746) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007413) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00492622) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00556462) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00764638) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00772926) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00499142) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00681172) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00566464) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00630734) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00654686) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003496) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008896) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002462) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003062) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009386) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004656) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00368028) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009348) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000697) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00064932) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00055094) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00223048) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00256848) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00044964) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00033968) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0007626) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00070684) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00284606) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00324824) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002516) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003152) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0138456) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0129777) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00059394) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00057078) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00017092) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00013866) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003646) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00005744) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0165639 #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.