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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 .00135556)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039494)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00238293)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00371605)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00590056)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00249587)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00199634)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00214913)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000430022)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000279194)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000277296)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167719)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00201495)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00265599)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00274959)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0017162)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233013)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00195156)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00216512)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00228778)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000762)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024502)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008088)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006858)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025422)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000691)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117196)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025834)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024976)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00028766)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000249374)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00078288)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000922996)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000158014)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000122912)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000248082)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024209)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000979388)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00112483)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006772)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007816)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010966)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001178)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00493713
#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 .00133668)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003792)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00238945)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00373768)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00591799)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00251408)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00202143)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00213787)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000430384)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000280764)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000282116)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .001724)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207712)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00266269)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00279688)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0130587)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00236456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00196783)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00216127)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231941)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007186)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023536)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000676)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007076)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024082)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006768)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00118317)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025652)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024832)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273766)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000248284)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000798152)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000918306)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000157814)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000122818)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000259264)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000243572)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0009775)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00111802)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006636)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007824)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00488366)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00437555)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000220544)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000217782)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00005243)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000052506)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008492)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000778)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00494965
#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 :