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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 .00733321)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000202507)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0107345)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0184421)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0278196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0130184)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0103313)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102102)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00185909)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00136966)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .001339)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00915422)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0105684)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0139097)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0141871)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00935169)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0127106)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .010523)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0115973)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .012475)   #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041766)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000119947)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003608)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035573)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000147527)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035667)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00603404)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000141933)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000115587)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00102197)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000989941)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00382865)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00450015)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007522)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000548186)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00133154)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00132921)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00529474)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00589776)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00005238)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000051066)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000063706)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000054786)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0262247
#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 .00736696)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000204386)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0107734)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0183617)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0278813)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .012956)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102447)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102379)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00185245)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00134786)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00137144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00913173)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0105307)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0138189)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0141937)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00932903)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0127235)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0105957)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0116851)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0123165)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041967)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000147334)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036274)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035313)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000137574)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036213)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00602775)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013278)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000115253)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00101516)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000987599)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00384011)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00449641)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000746613)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000525054)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00130705)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00135775)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00531363)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00594855)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036993)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039446)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .024962)   #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0232115)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00119396)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00115425)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000296994)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00027702)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040813)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003808)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0261819
#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 :