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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 .00103061)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036916)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0019125)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00321635)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00506081)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00226905)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00178693)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00182356)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000347743)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000231042)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000252846)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00144841)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166958)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00220513)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00228419)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00141252)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00197077)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0016383)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00181534)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193436)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009111)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037659)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007281)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009237)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029789)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006846)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00107558)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031122)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024041)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000216411)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000200935)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000757905)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000803832)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000149215)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000142372)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000211446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000195905)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000839567)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00096088)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009162)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010155)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011451)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012678)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00434299
#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 .00103682)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037725)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00191977)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0031786)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00504125)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00223974)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00184401)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00192171)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000350464)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000261621)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000237817)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00150513)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00172407)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0117384)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00226109)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00141607)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019311)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00158897)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00175542)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00186132)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007582)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030717)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006635)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010603)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029794)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006529)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00105098)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030216)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032972)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000215958)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000198274)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000697548)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000784333)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00013472)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000106865)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000238719)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000194223)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000818213)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000931682)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008948)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009931)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0041749)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00391847)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000213063)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000226687)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000041025)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000042813)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011041)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012605)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00461835
#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 :