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 .00111937) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000039512) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00212647) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .0033776) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00557473) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00253919) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00195767) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00206619) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000409402) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000232531) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000278018) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00156515) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00170744) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00221803) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00228588) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00142788) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00213471) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00179239) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00193856) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00203824) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000021856) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000028707) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008622) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011888) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000032698) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008606) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00107954) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00003323) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000025005) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000213893) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000211555) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000685541) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000842183) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000148048) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000121079) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000212001) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000215563) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000832915) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000966729) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007916) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010606) #primes = 8 #prunedViaCodim = 0 Strategy: IndependentSet (time .000011971) #primes = 9 #prunedViaCodim = 0 Strategy: IndependentSet (time .000013325) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00469141 #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 .00113516) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000035521) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00201999) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00337663) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00529933) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00225059) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00183588) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .00203989) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000410969) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000243238) #primes = 0 #prunedViaCodim = 0 Strategy: Factorization (time .000283543) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00161266) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00183165) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0114687) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00231802) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00160772) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00198418) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00173239) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .0018609) #primes = 0 #prunedViaCodim = 0 Strategy: Linear (time .00204585) #primes = 0 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010272) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000031274) #primes = 1 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007358) #primes = 2 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010983) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00002984) #primes = 3 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00000851) #primes = 4 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .00106341) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000033145) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000023597) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000214263) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000197888) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000699433) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000820682) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000135162) #primes = 6 #prunedViaCodim = 0 Strategy: Factorization (time .000108699) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000212154) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000197638) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000835932) #primes = 6 #prunedViaCodim = 0 Strategy: Linear (time .000953353) #primes = 6 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000007809) #primes = 7 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000010611) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .0044719) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .00401106) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000211764) #primes = 8 #prunedViaCodim = 0 Strategy: Birational (time .000210127) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000042044) #primes = 8 #prunedViaCodim = 0 Strategy: Linear (time .000038689) #primes = 8 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000008958) #primes = 9 #prunedViaCodim = 0 Strategy: DecomposeMonomials(time .000011273) #primes = 10 #prunedViaCodim = 0 Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00431261 #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.