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 .00128491) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000044597) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0022105) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00375601) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00575155) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00257144) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00209909) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00213441) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0004003) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000289939) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000261993) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170696) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00200994) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00258048) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00268269) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00169874) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235162) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00189852) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00214635) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00229308) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000872) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036345) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007261) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009093) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027591) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012474) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117498) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002909) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027847) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000242123) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000231727) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00079394) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000922935) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000162116) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000139803) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000242933) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000222329) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00100267) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00115984) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008191) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009432) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000016938) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000012938) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00502185 #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 .00126389) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000054459) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00219599) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00367431) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0058223) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00259293) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00208011) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00214974) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000428052) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000267516) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000274278) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00169562) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00197033) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00261459) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0027661) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00169097) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00233767) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00196082) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00221384) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00225202) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014573) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000034366) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007005) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009023) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027875) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008004) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00117908) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029641) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029977) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000229437) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000231886) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00080322) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000923811) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000171054) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000129039) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000246929) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000235971) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000988717) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00113997) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016806) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009988) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00474654) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00440447) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000242924) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000239793) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000049082) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000045607) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009219) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010282) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00493098 #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.