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 .0047244) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00016368) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00765474) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0128169) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0197096) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00908444) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00724378) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00681882) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00135272) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00094718) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00096168) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .006252) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00687524) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00903008) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00896102) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00631696) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00864794) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00721496) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00756098) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00786598) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004162) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011312) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002886) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003822) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001132) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003274) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00411228) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00011588) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008898) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0007507) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00059996) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00277926) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00290754) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00048514) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00036704) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00084614) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00073068) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00351556) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0040112) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003464) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003452) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000436) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .00004902) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0160435 #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 .00413474) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00013) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00678346) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0109596) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0172279) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00785626) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00647144) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00631878) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .001126) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0008241) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0007862) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00544738) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00607132) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0748677) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00826088) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0053156) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00774352) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00614996) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00673782) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00770918) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003446) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001206) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002708) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004082) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00012502) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000561) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00398274) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0001306) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00008134) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0007548) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00062056) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00253152) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00269024) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0004789) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00035992) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00082856) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00077496) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00315864) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00339776) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003206) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003646) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0158197) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0146577) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0007816) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00072484) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00015566) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0001472) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000362) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00004268) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0162078 #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.