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 .000970261) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026882) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00155187) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00258483) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00398742) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00180699) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00145971) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00150345) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00028236) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000198443) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000193396) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00177728) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00148957) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0019155) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00204252) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00127612) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00172115) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0014393) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00193475) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00171094) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006429) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018495) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006464) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005408) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019017) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005816) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000957011) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000020698) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017916) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000179575) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00015281) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000566146) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000661994) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000105989) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000085771) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000205926) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000180756) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000736697) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000837169) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006946) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007411) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000009992) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000007689) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00435958 #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 .00101796) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028665) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00181846) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00272419) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0243659) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00199922) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00160872) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0015866) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000292505) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000207352) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000199157) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00133062) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0016548) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00206168) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00213062) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00130753) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00181785) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156416) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00166799) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00174657) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005784) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017398) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007043) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005001) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001851) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004816) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000879184) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018454) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017479) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000166838) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000152244) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00057042) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000681413) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000112613) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000088692) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000233393) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000191391) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000833339) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000930151) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012374) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006102) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00412616) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00429837) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000164264) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000150426) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000044779) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000040048) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009345) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007203) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00444844 #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.