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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 .00237435)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000072982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0036958)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00631255)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0171989)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00462262)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00353752)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00358209)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00061298)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000459032)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000479188)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00306025)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00354109)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0046706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00475649)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00313711)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00420126)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00352973)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00388953)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00410904)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017184)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000048216)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000015448)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014146)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000048596)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001538)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00210734)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000047092)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000046076)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000376446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000330604)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0013306)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00153211)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00024446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000189968)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00047931)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000434316)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00176375)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00200253)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001424)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014822)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000024004)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000023162)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0104909
#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 .00244226)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000072636)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0038471)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00650502)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00983316)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00458305)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .003561)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00359456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000656156)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000486616)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000463036)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00303248)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00358083)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00469694)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00471943)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00312595)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00419428)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00350218)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00379905)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00405374)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017666)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000045662)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013876)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000162)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000054336)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016644)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00209533)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000472)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000043502)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036744)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000307578)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00126988)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00144488)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000237872)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00018466)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000422106)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00041866)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00170005)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00187956)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014882)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001365)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0082566)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00757646)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000345868)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000323364)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000099904)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000093726)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016092)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000015682)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0103854
#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 :