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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 .00131792)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003931)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00242944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00369228)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00585648)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00246382)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00196868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209171)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000422982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000272102)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00027396)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166812)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019965)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00260539)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00270071)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00168357)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .002294)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00191044)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00213999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0022866)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010312)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002529)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006574)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000677)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023838)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006622)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0011892)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023224)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024554)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00027337)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000249206)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000776298)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000929898)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00015563)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000121264)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000244808)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000238934)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00097239)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00111188)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006984)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006912)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001264)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011984)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00492326
#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 .00132163)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040208)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00240496)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00376814)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00586756)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00253456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0019947)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00211554)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00042689)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000303264)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000278076)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169597)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00208934)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0189682)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00281533)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00177783)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235011)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019264)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00219593)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00234901)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000975)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026028)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000836)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006864)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025638)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006914)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00124252)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002566)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040076)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00028879)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00024806)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000782298)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000920446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155142)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000119772)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000252472)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000241248)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00099185)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0011218)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000693)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007072)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00522056)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00460076)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000229048)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000217382)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000056876)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000068454)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009854)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008426)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00501784
#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 :