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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 .00191575)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000056144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00328601)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00523029)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00819592)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00354236)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00281477)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0298923)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000608122)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00037192)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000370598)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00259655)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00289763)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00375094)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00391632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00246409)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00333534)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00282632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00309355)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00324759)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012566)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041974)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011142)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010856)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037448)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010648)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00171297)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003508)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034962)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000357886)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000326266)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0011002)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00130142)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021721)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0001666)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000368994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000357368)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0014054)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00159796)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011394)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010454)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000018564)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000016992)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00825182
#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 .00195749)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000058198)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00336884)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00537316)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00831231)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00357763)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00286029)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00297795)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000577032)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000379204)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000382922)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00249184)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00292001)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00378456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00394318)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00245542)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00336769)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00277678)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0030223)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00322238)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001127)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000359)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011332)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001031)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033874)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010202)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00161527)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003553)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034776)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000348108)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000323172)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00107695)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00127441)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021032)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000158926)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000347604)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000342868)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0013838)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00157555)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010394)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010166)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00669324)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00611699)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000262624)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000260548)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000078638)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00007519)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011732)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010946)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0080812
#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 :