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 .00262607) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000079659) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00412562) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00660284) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00561885) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00386142) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00343819) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000719995) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000465358) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000491328) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00313717) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0031339) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00433207) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00436411) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00364937) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00347201) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00426282) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014299) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039444) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000013931) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019823) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00213618) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039193) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033738) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000442714) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .00138056) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .00161794) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000286172) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000173409) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00046006) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00178002) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00207989) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012376) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009749) #primes = 7 #prunedViaCodim = 0 Strategy: IndependentSet (time .000020761) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00720096 #minprimes=6 #computed=8 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 .00238278) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00007728) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00408591) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00776858) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00568578) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00348298) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00369434) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000610069) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000417902) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000401509) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00303264) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00308432) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0043809) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00461616) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00356519) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00399353) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00420443) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014331) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000045394) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011171) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000009746) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00212205) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000044046) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000042401) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000443688) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .00140825) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .00163449) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000268584) #primes = 5 #prunedViaCodim = 0 Strategy: Factorization (time .000199765) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .000471476) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00189225) #primes = 5 #prunedViaCodim = 0 Strategy: Linear (time .00213388) #primes = 5 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012592) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000012061) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .00864596) #primes = 7 #prunedViaCodim = 0 Strategy: Birational (time .000431489) #primes = 7 #prunedViaCodim = 0 Strategy: Linear (time .000084217) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000014365) #primes = 8 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .140531 #minprimes=6 #computed=8 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.