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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 .00101977)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034609)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00175671)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00293941)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00465291)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0280836)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00203016)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00195878)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000377773)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000262066)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000241063)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00132989)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00154851)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00202974)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207656)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00134702)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00177711)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00150136)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167722)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00178852)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008018)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025141)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006711)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007648)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024235)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008419)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000973753)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026092)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021604)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000197162)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000183565)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000632366)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000736535)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000127554)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000100898)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000207214)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000192716)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000803261)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000912166)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007872)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008599)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010582)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010434)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00458689
#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 .00105427)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033998)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00177386)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00298227)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00469254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00206973)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00166303)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00167885)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000322422)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000216777)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000220735)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00135123)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00159409)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211066)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215043)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00133338)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00181991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00153014)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167181)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00178349)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007356)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024066)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006669)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009128)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024204)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008033)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00094999)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022982)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020719)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000197479)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000182292)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000622761)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000725838)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000122209)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000102412)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000198582)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000187093)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000783444)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000893204)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009399)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009122)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00383781)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00346969)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000161882)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000151696)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035981)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035684)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008146)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008295)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0045629
#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 :