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 .00112244) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004146) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00201471) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00348359) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0141837) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00244394) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00204829) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00203869) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000414361) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00032989) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000311477) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00158378) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00171117) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227838) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00236874) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00149742) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00202327) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00171759) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00191605) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00209325) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010585) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032679) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008033) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011261) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000030463) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007896) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00120074) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028871) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025877) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00022653) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00021454) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000799847) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000923873) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000164292) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000137101) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000219539) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000210413) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000939935) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00106519) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000944) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010388) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000012243) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000013389) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00520243 #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 .00115986) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039926) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0020237) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00356176) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0056542) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00258604) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00211576) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00203728) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000448473) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000328727) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000339982) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00157847) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00179549) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0024629) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00239755) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .001488) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00205439) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00172317) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00193436) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00206846) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010145) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029746) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000818) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012574) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028568) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000081) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117982) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000030736) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026047) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000228585) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000217107) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000810742) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000930569) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000167155) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000138296) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000223791) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00021265) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000913206) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0011037) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000893) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000101) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0044319) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00399433) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000180595) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000206335) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000043625) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000042624) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010086) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011585) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00513637 #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.