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MinimalPrimes :: minprimes

minprimes -- minimal primes in a polynomial ring over a field

Synopsis

Description

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

Caveat

This will eventually be made to work over GF(q), and over other fields too.

Ways to use minprimes :