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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 .000970261)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00155187)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00258483)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00398742)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00180699)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00145971)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00150345)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00028236)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000198443)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000193396)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00177728)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00148957)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019155)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204252)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00127612)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00172115)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0014393)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193475)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00171094)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006429)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018495)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006464)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005408)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019017)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005816)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000957011)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020698)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017916)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000179575)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00015281)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000566146)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000661994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000105989)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000085771)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000205926)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000180756)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000736697)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000837169)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006946)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007411)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000009992)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000007689)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00435958
#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 .00101796)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028665)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00181846)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00272419)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0243659)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00199922)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00160872)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0015866)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000292505)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000207352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000199157)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00133062)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0016548)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00206168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00213062)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00130753)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00181785)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156416)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166799)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00174657)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005784)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017398)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007043)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005001)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001851)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004816)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000879184)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018454)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017479)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000166838)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000152244)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00057042)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000681413)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000112613)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000088692)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000233393)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000191391)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000833339)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000930151)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012374)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006102)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00412616)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00429837)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000164264)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000150426)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000044779)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000040048)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009345)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007203)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00444844
#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 :