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 .00101977) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034609) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00175671) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00293941) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00465291) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0280836) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00203016) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00195878) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000377773) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000262066) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000241063) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00132989) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00154851) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00202974) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207656) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00134702) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00177711) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00150136) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00167722) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00178852) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008018) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025141) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006711) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007648) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024235) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008419) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000973753) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026092) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021604) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000197162) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000183565) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000632366) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000736535) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000127554) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000100898) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000207214) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000192716) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000803261) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000912166) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007872) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008599) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010582) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010434) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00458689 #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 .00105427) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033998) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00177386) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00298227) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00469254) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00206973) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00166303) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00167885) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000322422) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000216777) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000220735) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00135123) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00159409) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00211066) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215043) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00133338) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00181991) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00153014) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00167181) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00178349) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007356) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024066) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006669) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009128) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024204) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008033) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00094999) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022982) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000020719) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000197479) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000182292) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000622761) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000725838) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000122209) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000102412) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000198582) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000187093) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000783444) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000893204) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009399) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009122) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00383781) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00346969) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000161882) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000151696) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000035981) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000035684) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008146) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008295) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0045629 #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.