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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 .00111763)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039576)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .001931)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00334532)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00529439)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00236732)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00192485)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00188938)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040353)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000232677)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000224397)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00150279)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00185685)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233575)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00241623)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00149869)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00219828)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00180456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00209115)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211003)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009378)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030293)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007543)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012798)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003204)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008697)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0012243)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000047456)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025154)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000216998)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000210394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000698095)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0008751)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000149959)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000126162)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000218879)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00023118)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000862243)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00105743)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008574)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001187)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000016727)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000014634)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00521472
#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 .00119142)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038797)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0109857)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00324254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0051311)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00223147)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0017754)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00182648)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000352196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000246661)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000226433)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156558)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170409)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00232158)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00236181)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00149446)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00205232)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00174154)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00184656)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00196574)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010526)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031954)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001039)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001216)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038787)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008927)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00121042)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041058)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025445)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000246109)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000226864)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000721909)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000827019)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000136477)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000131056)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000239766)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000203455)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00101032)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00102444)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008646)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012638)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00441738)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00415718)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000203144)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000195321)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000046571)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000041826)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010582)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013232)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00536551
#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 :