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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 .00124982)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000043991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00231031)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00388945)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00614678)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00268747)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00213876)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00229481)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00044339)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000284456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000277806)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00180816)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00201314)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00285661)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00267622)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00171749)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227199)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00195491)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211076)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0023058)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008634)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004612)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007607)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022211)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028799)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013131)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00121361)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002994)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031043)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000236374)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000224601)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000790596)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00090803)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000193624)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000128601)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000243847)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000231405)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000985015)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00110274)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008256)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009917)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001776)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000013807)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0049409
#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 .00122098)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000044991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00240125)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00381604)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00631783)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00279546)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00230889)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00227349)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00041532)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000280071)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000299363)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00177696)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00219448)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00273075)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0028175)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00182124)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252337)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204823)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0023007)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00225506)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016313)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035006)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007013)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009815)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031014)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027744)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0013224)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031433)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029913)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000269336)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000243752)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000845091)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00117631)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00018641)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000129532)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000249794)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000239835)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00110635)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00129998)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008042)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022648)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00511982)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00440897)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000260016)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000228711)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000051386)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00006688)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009557)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011149)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00505529
#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 :