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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 .00111937)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039512)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00212647)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0033776)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00557473)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00253919)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00195767)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00206619)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000409402)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000232531)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000278018)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156515)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170744)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221803)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00228588)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00142788)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00213471)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00179239)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193856)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00203824)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021856)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028707)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008622)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011888)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032698)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008606)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00107954)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003323)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025005)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000213893)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000211555)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000685541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000842183)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000148048)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000121079)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000212001)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000215563)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000832915)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000966729)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007916)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010606)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011971)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000013325)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00469141
#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 .00113516)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035521)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00201999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00337663)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00529933)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00225059)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00183588)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00203989)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000410969)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000243238)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000283543)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00161266)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00183165)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0114687)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231802)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00160772)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00198418)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173239)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018609)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204585)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010272)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031274)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007358)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010983)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002984)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000851)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00106341)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033145)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023597)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000214263)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000197888)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000699433)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000820682)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000135162)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000108699)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000212154)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000197638)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000835932)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000953353)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007809)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010611)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0044719)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00401106)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000211764)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000210127)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000042044)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000038689)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008958)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011273)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00431261
#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 :