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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 .00151196)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004981)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00256228)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00431685)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00667584)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00301758)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0023873)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00244059)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000458674)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000322695)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000302576)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00200741)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00232524)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00306548)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00315293)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00202123)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00274157)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227451)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00250187)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0027006)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009153)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034102)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008037)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008742)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028251)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013192)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00136039)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028805)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034946)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000263639)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000258723)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000910672)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00106209)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000185541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000147965)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000286161)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000267019)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00118089)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00132026)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008711)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008979)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000019541)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000013327)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00581203
#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 .00151303)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000048755)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00256271)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00429515)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00669776)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00302547)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00239873)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00242844)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000466988)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00031607)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000308141)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00200323)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00232144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00306999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00315968)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0020141)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00275685)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00229131)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00253063)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00266452)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013555)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033328)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008076)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010446)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002833)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008543)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00135627)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029061)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026086)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000263264)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000255138)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000917863)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00106315)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000181663)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000149235)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000283084)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000279439)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0011677)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00133353)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008435)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000916)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00549334)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00517055)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000276998)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000255225)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000060439)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000055023)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000921)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009837)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00580812
#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 :