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 .00221745) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000782) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00355953) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0062854) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00954073) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0045876) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0729112) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00366628) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000652475) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000523525) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0005405) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00302068) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0035258) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00466815) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0046373) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00313317) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00414545) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00347818) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00390755) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00407065) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002205) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000675) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019075) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022875) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000065075) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019475) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0022342) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000683) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000059025) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00037895) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0003626) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0014054) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00163313) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0002814) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000235325) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000443375) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0004401) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00186117) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00203357) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002525) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000028025) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000028525) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0107779 #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 .0023909) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000080675) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00377058) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00647168) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00989383) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00472765) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00371283) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00370637) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0006537) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000496425) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000493825) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00306688) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0036077) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0046591) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00481555) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00302485) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00414643) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00349958) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00376962) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00399613) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000022) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000066675) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002285) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000608) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001905) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00217523) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000065275) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000541) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000364375) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00035165) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00133482) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .0015702) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000273775) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000229825) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000434575) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00043385) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .0018972) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00192155) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002075) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023375) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00839215) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00771125) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00037345) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0003591) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0000878) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .0000854) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002245) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000026675) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0105054 #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.