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 .00113316) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000037168) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00194341) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00377279) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00559896) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00250195) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00198295) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00201429) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000369494) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000271176) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000251129) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00161623) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00183032) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00238921) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00255786) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00160527) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00230951) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0019324) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00210993) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215121) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012034) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000040274) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008331) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000024355) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033231) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008334) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00128423) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032764) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027122) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000293527) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000242434) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000853533) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000936349) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000153343) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000132536) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000243708) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000251984) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000988454) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00108885) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009023) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011242) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .0000177) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000014553) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0059663 #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 .00115003) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000057238) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0100164) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00397765) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00589601) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00269868) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00226939) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00228739) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000468481) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000285426) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000274328) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00188639) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00211956) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00285769) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00291016) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00178875) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .002408) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00207907) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235583) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00250063) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000016817) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000068357) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010812) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013692) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000036054) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022674) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00135079) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003603) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000027982) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000263023) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000224541) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00084863) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000996456) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000185641) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000146339) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000299773) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000269931) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00101389) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00116392) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001322) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0057183) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00493721) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000224906) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000249685) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .00005115) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0000446) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011759) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013273) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00643017 #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.