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 .00132364) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039194) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00238708) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00376091) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00587589) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00250082) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00199868) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00209827) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000425712) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000276718) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000273652) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00167704) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00201131) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00263599) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00268941) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00168849) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231327) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00194293) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00212706) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227293) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000761) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025994) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006666) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007022) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002476) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006744) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117667) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025182) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022166) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000267794) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000249756) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000775604) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000917622) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000155182) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000122752) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024711) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00024338) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00098293) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00113749) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006892) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006878) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .00001107) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010696) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0049138 #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 .0013254) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038084) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00239289) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00371984) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00591313) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00249489) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00200862) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00214129) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000429122) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000301806) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000276742) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00172406) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00204009) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00269437) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0157168) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170687) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00238886) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00194704) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0021378) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227849) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008012) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025086) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007762) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007918) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023576) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007664) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00119408) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023888) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021748) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000268284) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000250998) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000792254) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000917848) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000155908) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00011817) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0002537) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000246026) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00100162) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00111831) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006838) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008148) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00487324) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00443477) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000224734) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000219932) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000055146) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000050856) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008132) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007536) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00499712 #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.