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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 .00308366)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000096068)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00518354)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00842258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0131961)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00582577)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00466562)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0492834)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00092851)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00061256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000604344)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00401204)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00481899)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00630109)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00647515)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00415254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0056556)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00457196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00512961)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00533779)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000204)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006347)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018106)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019362)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000059656)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019252)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00273217)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000062256)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000062208)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000534768)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000507904)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00177824)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00212754)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000346298)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000266704)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0005939)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00056683)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00229541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00274324)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019812)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019084)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000032812)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000030904)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0135455
#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 .00311022)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000099138)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00519617)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00869196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0133767)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00595231)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00468223)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00477198)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000893336)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000618868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000602466)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00399347)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00476494)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00620901)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00639509)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00416202)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00573624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00459145)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00505101)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00521414)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020492)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000063038)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019132)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019404)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000060702)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018836)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00274224)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000063212)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000058688)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00054675)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00050343)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00173086)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209183)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000335856)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000258402)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000576954)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000563794)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00226115)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00258659)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019536)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021324)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0110169)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00996639)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000444266)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000441472)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014604)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000133966)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023224)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020152)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0132508
#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 :