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 .00237435) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000072982) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0036958) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00631255) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0171989) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00462262) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00353752) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00358209) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00061298) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000459032) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000479188) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00306025) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00354109) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0046706) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00475649) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00313711) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00420126) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00352973) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00388953) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00410904) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017184) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000048216) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000015448) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014146) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000048596) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001538) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00210734) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000047092) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000046076) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000376446) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000330604) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0013306) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00153211) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00024446) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000189968) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00047931) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000434316) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00176375) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00200253) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001424) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014822) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000024004) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000023162) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0104909 #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 .00244226) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000072636) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0038471) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00650502) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00983316) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00458305) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .003561) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00359456) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000656156) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000486616) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000463036) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00303248) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00358083) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00469694) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00471943) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00312595) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00419428) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00350218) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00379905) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00405374) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000017666) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000045662) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013876) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000162) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000054336) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016644) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00209533) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000472) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000043502) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00036744) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000307578) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00126988) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00144488) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000237872) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00018466) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000422106) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00041866) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00170005) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00187956) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014882) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001365) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0082566) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00757646) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000345868) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000323364) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000099904) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000093726) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016092) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000015682) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0103854 #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.