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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 .00130632)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000044051)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00246114)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00415575)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00620632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00278763)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00227985)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00229994)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000430413)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000330674)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00027706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193307)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211645)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00277901)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00286846)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00179186)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0024923)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00199624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233972)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00268616)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010861)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000049366)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008951)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009769)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000045598)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002256)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00135102)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030379)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000047403)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000298007)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000222105)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000855351)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00098002)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000156102)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000171585)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000241284)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000233142)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00102124)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00113974)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000906)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010317)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000019229)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000014391)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00518078
#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 .00127027)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000047096)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00243745)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00398942)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00665377)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .003002)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00227055)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00247792)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000464564)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000337157)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000290357)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00186061)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215782)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00295837)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00316107)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00223772)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00259912)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00213155)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227445)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018532)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036173)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006969)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009699)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029611)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007995)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00133792)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039753)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031632)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000277968)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000239765)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000857302)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000967595)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000200301)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000149992)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00030052)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000227677)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00109052)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00130943)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008118)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010275)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00582127)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00545789)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000288326)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000261796)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000056046)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000050406)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001028)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010869)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00540079
#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 :