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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 .0012783)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040174)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00229866)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00355522)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00557342)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00237626)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00188484)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00198938)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040198)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000260146)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000256816)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00161639)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193224)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252046)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0025917)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00162345)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00222804)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00185259)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00206104)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00217006)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010828)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002201)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006132)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007294)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022628)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007122)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0011764)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022788)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024216)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000275744)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000234582)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00073849)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000866792)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155398)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000112764)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000238776)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000225216)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00093057)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00106628)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006522)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006224)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010016)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000009304)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00469175
#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 .00128675)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000371)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00230211)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00355153)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00559135)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00239809)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00190099)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00200474)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040927)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000260986)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000260504)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0016651)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00196371)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .017435)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00260171)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167222)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221408)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00185094)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0020726)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0022019)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009296)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023832)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005976)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006938)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023236)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006002)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .001166)   #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022618)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022114)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000263866)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000235104)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000745258)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000870788)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00014508)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000113132)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024096)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000229218)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00095109)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00106873)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006824)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000721)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00479681)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00428555)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00020143)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000214058)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000053596)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000048374)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009532)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006996)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00478281
#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 :