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 .00113963) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000038343) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00204452) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00337144) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00533287) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00233296) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00187435) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00192847) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000382899) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000253043) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000248385) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00152885) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0017747) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00234508) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00241933) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00151507) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207241) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173575) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00192133) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0020411) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007582) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027658) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007158) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008371) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027386) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007329) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00106833) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026827) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022897) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00022959) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000217966) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000725366) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000842205) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000141466) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000117088) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000227585) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000214224) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000896147) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00103169) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000766) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008474) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000010831) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011287) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00442862 #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 .00115338) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036357) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00204569) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00340192) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00536595) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00236765) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0019195) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00197376) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000384614) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000268657) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000258226) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156208) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0018341) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0095883) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00264934) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170898) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00227442) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .001887) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00210969) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00224518) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009246) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029734) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007377) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009143) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026416) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007907) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00116281) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028991) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002397) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000242965) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000229498) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000775653) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000909515) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000153748) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00012457) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000243942) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000241392) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000968241) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00112629) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007546) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008819) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0047505) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00439698) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000227807) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00022433) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000048627) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000045241) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000868) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009221) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00488495 #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.