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 .00104664) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000029343) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00165512) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00274854) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00425941) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00192544) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00151484) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00156715) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00029942) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000211169) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000205529) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00145262) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00159112) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00205248) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00215336) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00137126) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00185624) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00154454) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0017112) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00180827) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007202) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00001978) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004759) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007075) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000018911) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004861) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000904332) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019156) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019581) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000174104) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000165632) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000597175) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000702709) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000117241) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .00009) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000199909) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000197619) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000801902) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000914519) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006636) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005202) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000008788) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000007785) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00443888 #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 .00108738) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000031885) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0017157) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00285035) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0047945) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0020319) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00157617) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00162703) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000303032) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000213779) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000210855) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00138165) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00160253) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00235824) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00242003) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00150996) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00183934) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00153113) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170914) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00180688) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007777) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019096) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000006102) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007584) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019199) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000004946) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000931019) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021216) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000019203) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000187776) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000163158) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000597835) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000701215) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000113298) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000089919) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000203251) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .00018674) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000779839) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000875452) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005897) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .0000055) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00376889) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00330968) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00014226) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000135494) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000037734) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000037965) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000578) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000005783) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00430701 #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 |
This will eventually be made to work over GF(q), and over other fields too.