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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 .00128491)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000044597)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0022105)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00375601)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00575155)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00257144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209909)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00213441)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0004003)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000289939)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000261993)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170696)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00200994)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00258048)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00268269)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169874)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235162)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00189852)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00214635)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00229308)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000872)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036345)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007261)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009093)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027591)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012474)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117498)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002909)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027847)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000242123)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000231727)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00079394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000922935)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000162116)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000139803)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000242933)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000222329)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00100267)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00115984)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008191)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009432)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000016938)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012938)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00502185
#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 .00126389)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000054459)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00219599)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00367431)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0058223)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00259293)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00208011)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00214974)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000428052)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000267516)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000274278)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169562)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00197033)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00261459)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0027661)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169097)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233767)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00196082)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221384)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00225202)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014573)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034366)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007005)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009023)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027875)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008004)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117908)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029641)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029977)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000229437)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000231886)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00080322)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000923811)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000171054)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000129039)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000246929)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000235971)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000988717)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00113997)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016806)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009988)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00474654)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00440447)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000242924)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000239793)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000049082)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000045607)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009219)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010282)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00493098
#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 :