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 .00378324) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00015446) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00611026) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0102754) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0156271) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0073277) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00580412) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00582468) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00103152) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007432) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00074146) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00494304) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00563318) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00739982) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00761682) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00499718) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0067757) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00564666) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .006217) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00655534) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002838) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009862) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000271) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003022) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009382) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002332) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00356926) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000094) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007424) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0006258) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0005856) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00222024) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00255874) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00045282) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00033862) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00073968) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00070502) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00284076) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00319836) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002488) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003098) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00003788) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004518) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0147092 #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 .00373932) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00014748) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00603736) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0102981) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0157183) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00741392) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0058329) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00591928) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00103558) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000814) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00070618) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0050851) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0057588) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .067557) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00790436) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0050344) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0069706) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00572268) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00631998) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00669126) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002924) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00009616) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002478) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003018) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008994) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003072) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00353756) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00010278) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000077) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00062082) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00057334) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00223124) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0025814) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00044644) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0003802) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00075362) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00067732) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00288204) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0032536) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002702) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003132) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0144311) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0132566) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00070688) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0006936) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00014846) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00013964) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003538) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00006276) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0153268 #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.