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 .0036191) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00013188) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0063888) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0105254) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0165122) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00751808) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0058209) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00593674) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00101906) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00074828) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00072248) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00531016) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00596126) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00764086) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00762142) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00500092) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0069678) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00573954) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0062608) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00666456) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003134) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011408) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003104) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003894) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011358) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002938) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0037232) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011136) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007794) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00066664) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00056238) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00234278) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00278328) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00043488) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00036628) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0007433) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00065956) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00295874) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00316474) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00005402) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003182) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00003716) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000418) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0146311 #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 .00372434) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001212) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00611902) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .010564) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0161743) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00739286) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00598944) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0057886) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00106652) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00085248) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00085512) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00528868) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00571378) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00795626) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0702051) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00506216) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00675254) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00564046) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00615398) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00650658) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003216) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010976) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002322) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004042) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00012474) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00006816) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0037994) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011884) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000791) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0006928) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00055586) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00234744) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00255324) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00042942) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0003429) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00076558) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00067694) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00289242) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00318802) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003046) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003444) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .014294) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0130038) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00070914) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00066972) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014964) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0001338) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003476) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003672) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0149747 #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.