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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 .00196867)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00005977)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00337519)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00528135)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0081821)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00359291)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00281373)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00294609)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00056747)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000377138)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000395632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00243711)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .002881)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00380236)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00388092)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00248453)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00339873)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0027997)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00308557)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0033252)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001617)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035374)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009492)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009576)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033164)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009526)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00173034)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033928)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032898)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000370734)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000326514)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00109029)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00128306)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000211272)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000162602)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000358208)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00035014)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00141094)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00160105)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010144)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009534)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000017884)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000015276)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00718151
#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 .00194264)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000057896)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00336378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0053316)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00825004)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00359493)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0028788)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0029825)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00056973)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00037947)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00038005)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00248293)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00294903)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00379752)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00396887)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .021242)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00343957)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00285261)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00311717)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00331236)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012794)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037544)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001094)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000962)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033494)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000951)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00173282)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035744)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034194)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000367378)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000323516)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00109157)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00132413)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000212546)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000160846)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000374996)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00035043)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00141392)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00159517)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001044)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001084)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00718587)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00642638)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000317192)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000316358)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000083562)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000077732)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001445)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010694)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00723709
#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 :