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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 .0047244)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00016368)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00765474)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0128169)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0197096)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00908444)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00724378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00681882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00135272)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00094718)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00096168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .006252)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00687524)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00903008)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00896102)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00631696)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00864794)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00721496)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00756098)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00786598)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004162)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011312)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002886)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003822)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001132)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003274)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00411228)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011588)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008898)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007507)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00059996)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00277926)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00290754)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00048514)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036704)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00084614)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00073068)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00351556)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0040112)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003464)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003452)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000436)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004902)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0160435
#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 .00413474)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013)    #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00678346)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0109596)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0172279)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00785626)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00647144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00631878)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .001126)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0008241)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007862)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00544738)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00607132)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0748677)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00826088)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0053156)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00774352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00614996)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00673782)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00770918)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003446)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001206)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002708)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004082)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012502)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000561)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00398274)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001306)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008134)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007548)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00062056)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00253152)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00269024)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0004789)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00035992)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00082856)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00077496)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00315864)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00339776)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003206)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003646)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0158197)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0146577)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0007816)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00072484)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00015566)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0001472)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000362)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004268)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0162078
#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 :