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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 .000914278)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002655)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00146201)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00241247)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00375189)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00167616)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00132793)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00136906)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000262788)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000183333)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000186352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00115279)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00134376)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00178581)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00182192)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00116906)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00158622)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00132523)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00146219)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156272)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005499)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016014)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000469)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004874)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017215)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005747)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000758419)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018086)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000015403)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000152099)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000142957)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000504553)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000600887)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000098393)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000079548)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000171018)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000165875)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000674446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000763511)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004862)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004947)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000007501)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000007196)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00382761
#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 .000930312)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027351)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00145734)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00249734)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0189884)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00174034)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00136771)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00142159)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000268364)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000188604)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000190759)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00121632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0014191)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00184691)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00189868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00120818)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00165346)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0013775)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00151908)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00163157)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005015)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018195)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004637)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004854)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017592)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004785)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000819552)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018951)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019756)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155245)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000145092)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000530541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000626796)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000102521)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000080316)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000175324)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000177582)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000707461)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00079415)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004974)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004945)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00331286)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00321745)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000149463)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00013334)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035138)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035066)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006479)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005542)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .004056
#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

Caveat

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

Ways to use minprimes :