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 .00196884) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000056762) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00325132) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00534673) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00823544) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0036095) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00282227) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00298118) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000556686) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000389524) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000378604) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00246018) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00292771) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00385903) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00391611) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0025365) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00343485) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00281803) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00315882) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00334542) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013108) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033058) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010134) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010306) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035046) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011666) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00165279) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036088) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033966) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00035114) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00032039) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0010812) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00129937) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000210392) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000163168) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000358472) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00035718) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00143016) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0016414) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001035) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001022) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000017554) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000016244) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00842719 #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 .00197907) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000056028) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00329183) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0158456) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0082457) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00359731) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00282062) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00298301) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00056629) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000382178) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000382088) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00248824) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00294115) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00383882) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00392544) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00253829) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00338681) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00283338) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00315115) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00332796) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012886) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003418) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011224) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010158) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033362) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011152) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00166692) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003599) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034718) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00034157) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000318406) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00108836) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00134005) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021153) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000161434) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00036218) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000357474) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00144456) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00169277) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011348) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010962) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0069237) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00641259) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00027727) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000272588) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000084168) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000082) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001256) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010792) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00847548 #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.