Re: plotting groups of polynomial roots

*To*: mathgroup at smc.vnet.net*Subject*: [mg51272] Re: plotting groups of polynomial roots*From*: mathma18 at hotmail.com (Narasimham G.L.)*Date*: Mon, 11 Oct 2004 01:25:31 -0400 (EDT)*References*: <ckajb5$m4j$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Roger Bagula <tftn at earthlink.net> wrote in message news:<ckajb5$m4j$1 at smc.vnet.net>... > If you take the first and last term away from a binomial polynomial and > set the result equal to zero, > you get a number of strange roots. > This method allows you to plot such roots. > I didn't know it would work when I wrote it up, > but I plan to use it in the future > on some other polynomial root structures. > > (* root group where x^q+1=(x+1)^q: binomial expansion without x^q and 1*) > digits=21 > s[q_]=Sum[(q!/((q-k)!*k!))*x^(q-k),{k,1,q-1}] > ExpandAll[s[2]] > ExpandAll[s[3]] > a=Flatten[Table[x/. NSolve[s[n]==0,x],{n,2,digits}]]; > a0=Floor[Abs[a]] > Dimensions[a][[1]] continued... b = Table[{Re[a[[n]]], Im[a[[n]]]}, {n, 1, Dimensions[a][[1]]}]; ChopEnds = ListPlot[b, PlotRange -> {{-1, 2}, {-1, 1}}]; central = ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 2 Pi}]; displ = ParametricPlot[{Cos[t] - 1, Sin[t]}, {t, 0, 2 Pi}]; Show[ChopEnds, central, displ]; Hi Roger, The roots are neatly herded onto unit circles centered on (0,0) and (-1,0), (except one point (-0.5, +/-1), as may be expected for complex roots of z^(1/n),(z+1)^(1/n) somehow with only negative real parts, |x|<1 . Regards. Nara