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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 .00108469)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003309)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00190363)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00312723)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00463167)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00205675)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00163469)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00177566)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000322062)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000222861)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000219298)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0014035)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00164365)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00216616)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00222843)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00143406)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019416)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00160753)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00185192)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019012)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006655)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019565)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005384)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005691)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020713)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006427)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000938813)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021532)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020137)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000191672)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000179083)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000633249)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000737306)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000120502)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000090458)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00020623)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000195835)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000823845)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00094064)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006952)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005884)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010152)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000008974)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00399425
#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 .00110534)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032286)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00181019)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00299719)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00468262)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00218475)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00171445)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00176303)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000325184)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000224161)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000218972)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00143488)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167803)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00219313)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00225245)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0203586)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0020065)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00159149)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00174707)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00188695)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005812)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002059)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005471)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007524)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021913)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005451)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00092725)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021801)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018531)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000194768)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000183501)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000615994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000719375)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000119237)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000091857)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000206979)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000198485)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00081307)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000925251)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007057)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006041)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00383416)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00364188)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000180675)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000176729)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000044735)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000040166)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006974)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009453)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00401381
#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 :