/* finding Misiurewicz points c=M(k,n) for fc(z)=z*z+c using Maxima CAS theory : The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653 Adam Majewski fraktal.republika.pl 2009.12.4 - 6 */ f(z,c):=z*z+c; fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); /* critical polynomial */ Q(p,x) :=fn(p,0,x); /* Misiurewicz polynomial */ MP(k,n,x):= Q(k+n,x) - Q(k,x); /* find Misiurewicz points among roots of Misurewicz polynomial but between above roots are other types of points */ /* M(k,n):=float(rectform(solve(MP(k,n,x),x))); fails for M(4,1) M(k,n) := allroots(%i*MP(k,n,x)); fails for MP(7,1) find MPI(k,n,x) which is irreducible divisor of MP(k,n,x) get list of coefficients of MPI = c_list:give_coefficients(P,x); find its greatest coefficient = c_max:lmax(c_list); find its binary size = log2(c_max) set fpprec>log2(c_max) map('float,bfallroots(%i*a)); */ F:-1.401155; /* plot critical curves for Misiurewcz points */ plot_M(k,n):=plot2d([Q(k+n,x),Q(k,x)],[x,-2.0,F]); /* gives a list of coefficients */ give_coefficients(P,x):=block ( P:expand(P), /* if not expanded */ degree:hipow(P,x), a:makelist(coeff(P,x,degree-i),i,0,degree)); log2(x) := log(x) / log(2); /* gives binary size of greatest coefficient */ Give_prec(P,x):= ceiling(log2(lmax(give_coefficients(P,x)))); /* examples : %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --------------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%% period 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% ------------------------------------------------------------------ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MP(2,1,x) = (x^2+x)^2-x^2 = x^3* /* x^(k+n) = center of period 1 */ (x+2) /* MPI(2,1,x) */ roots(MP(2,1)) = [x=-2.0,x=0.0] MPI(2,1,x) = (x+2) M(2,1) = roots(MPI(2,1)) = -2 --------------------------------------------------------- MP(3,1,x) = x^(3+1) * MPI(2,1,x) * MPI(3,1,x) ((x^2+x)^2+x)^2-(x^2+x)^2 = x^4* /* x^(k+n) = center of period 1 */ (x+2)* /* MPI(2,1,x) */ (x^3+2*x^2+2*x+2) /* MPI(3,1,x) */ roots( MP(3,1,x)) = [x=-1.115142508039935*%i-0.22815549365396, x=1.115142508039935*%i-0.22815549365396, x=-1.543689012692072, x=-2.0,x=0.0] MPI(3,1,x) = (x^3+2*x^2+2*x+2) M(3,1) = roots(MPI(3,1)) = allroots(x^3+2*x^2+2*x+2); [x=1.115142508039937*%i-0.22815549365396, x=-1.115142508039937*%i-0.22815549365396, x=-1.543689012692076] ------------------------------------------------- MP(4,1,x) = x^(4+1) * MPI(2,1,x) * MPI(3,1,x) * MPI(4,1,x) (((x^2+x)^2+x)^2+x)^2-((x^2+x)^2+x)^2 = x^5* /* x^(k+n) = center of period 1 */ (x+2)* /* MPI(2,1,x) */ (x^3+2*x^2+2*x+2)* /* MPI(3,1,x) */ (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2) /* MPI(4,1,x) */ roots( MP(4,1,x)) = [ [x=0.0, x=0.0, x=0.0, x=0.0, x=0.0, x=0.700620020235*%i+0.34390699597257, x=0.34390699597257-0.700620020235*%i, x=0.95628651080914*%i-0.10109636384562, x=-0.95628651080914*%i-0.10109636384562, x=-1.89291098790517, x=1.115142508039936*%i-0.22815549365396, x=-1.115142508039936*%i-0.22815549365396, x=0.44185160573511*%i-1.296355138172952, x=-0.44185160573511*%i-1.296355138172952, x=-1.543689012693117, x=-2.000000000001776] MPI(4,1,x) = x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2 M(4,1) = roots(MPI(4,1,x)) = [ x=0.700620020235*%i+0.34390699597257, x=0.34390699597257-0.700620020235*%i, x=0.95628651080914*%i-0.10109636384562, x=-0.95628651080914*%i-0.10109636384562, x=0.4418516057352*%i-1.296355138173036, x=-0.4418516057352*%i-1.296355138173036, x=-1.892910987907821] ------------------------------------------------------------- factor(MP(5,1,x)) = x^(5+1)*MPI(2,1)*MPI(3,1)*MPI(4,1)*MPI(5,1) x^6* (x+2)* (x^3+2*x^2+2*x+2)* (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* (x^15+8*x^14+28*x^13+60*x^12+94*x^11+116*x^10+114*x^9+94*x^8+70*x^7+48*x^6+32*x^5+20*x^4+10*x^3+4*x^2+2*x+2) MPI(5,1,x) = x^15+8*x^14+28*x^13+60*x^12+94*x^11+116*x^10+114*x^9+94*x^8+70*x^7+48*x^6+32*x^5+20*x^4+10*x^3+4*x^2+2*x+2 M(5,1) = roots(MPI(5,1,x)) = [x=0.59153377326145*%i+0.36636298342276, x=0.36636298342276-0.59153377326145*%i, x=0.40993310830098*%i+0.44455687925504, x=0.44455687925504-0.40993310830098*%i, x=0.68503129708367*%i-0.63675434658238, x=-0.68503129708367*%i-0.63675434658238, x=1.037652343793212*%i-0.017187977338355, x=-1.037652343793212*%i-0.017187977338355, x=1.086624831861381*%i-0.17589070597345, x=-1.086624831861381*%i-0.17589070597345, x=0.31688226388712*%i-1.222863787129814, x=-0.31688226388712*%i-1.222863787129814, x=0.066161509083841*%i-1.771257023355398, x=-0.066161509083841*%i-1.771257023355398, x=-1.973932044596817] ----------------------------------------------- factor(MP(6,1,x)); = x^7* (x+2)* (x^3+2*x^2+2*x+2)* (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* (x^15+8*x^14+28*x^13+60*x^12+94*x^11+116*x^10+114*x^9+94*x^8+70*x^7+48*x^6+32*x^5+20*x^4+10*x^3+4*x^2+2*x+2)*(x^31+16*x^30+120*x^29+568*x^28+1932*x^27+5096*x^26+10948*x^25+19788*x^24+30782*x^23+41944*x^22+50788*x^21+55308*x^20+54746*x^19+49700*x^18+41658*x^17+32398*x^16+23462*x^15+15872*x^14+10096*x^13+6096*x^12+3528*x^11+1976*x^10+1072*x^9+564*x^8+290*x^7+144*x^6+68*x^5+28*x^4+10*x^3+4*x^2+2*x+2) MPI(6,1,x)= (x^31+16*x^30+120*x^29+568*x^28+1932*x^27+5096*x^26+10948*x^25+19788*x^24+30782*x^23+41944*x^22+50788*x^21+55308*x^20+54746*x^19+49700*x^18+41658*x^17+32398*x^16+23462*x^15+15872*x^14+10096*x^13+6096*x^12+3528*x^11+1976*x^10+1072*x^9+564*x^8+290*x^7+144*x^6+68*x^5+28*x^4+10*x^3+4*x^2+2*x+2) allroots(MPI(6,1,x)) = [x=0.25054164301809*%i+0.43820113569629, x=0.43820113569629-0.25054164301809*%i, x=0.34189208433812*%i+0.43792424135946, x=0.43792424135946-0.34189208433812*%i, x=-1.818671833985708, x=0.62412663926224*%i+0.41300665288819, x=0.41300665288819-0.62412663926224*%i, x=0.67144813917988*%i+0.36985524536242, x=0.36985524536242-0.67144813917988*%i, x=0.64281646053538*%i-0.56220797009481, x=-0.64281646053538*%i-0.56220797009481, x=0.87336108665387*%i+0.025654588804545, x=0.025654588804545-0.87336108665387*%i, x=0.84210453983125*%i-0.30648214255773, x=-0.84210453983125*%i-0.30648214255773, x=0.39533838646742*%i-1.031069809935635, x=-0.39533838646742*%i-1.031069809935635, x=1.113778475827157*%i-0.15877652719083, x=-1.113778475827157*%i-0.15877652719083, x=-1.697495081160805, x=1.006001268931295*%i-0.016604586086526, x=-1.006001268931295*%i-0.016604586086526, x=1.113085141272325*%i-0.20968200039812, x=-1.113085141272325*%i-0.20968200039812, x=0.41516541987625*%i-1.274713379992677, x=-0.41516541987625*%i-1.274713379992677, x=0.16244422569328*%i-1.428671130759281, x=-0.16244422569328*%i-1.428671130759281, x=0.0075741248512867*%i-1.941582446689324, x=-0.0075741248512867*%i-1.941582446689324, x=-1.993536825665417] ---------------------------------------------- (%i12) factor(MP(7,1,x)); x^8* (x+2)* (x^3+2*x^2+2*x+2)* (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* (x^15+8*x^14+28*x^13+60*x^12+94*x^11+116*x^10+114*x^9+94*x^8+70*x^7+48*x^6+32*x^5+20*x^4+10*x^3+4*x^2+2*x+2)*(x^31+16*x^30+120*x^29+568*x^28+1932*x^27+5096*x^26+10948*x^25+19788*x^24+30782*x^23+41944*x^22+50788*x^21+55308*x^20+54746*x^19+49700*x^18+41658*x^17+32398*x^16+23462*x^15+15872*x^14+10096*x^13+6096*x^12+3528*x^11+1976*x^10+1072*x^9+564*x^8+290*x^7+144*x^6+68*x^5+28*x^4+10*x^3+4*x^2+2*x+2)* (x^63+32*x^62+496*x^61+4976*x^60+36440*x^59+208336*x^58+971272*x^57+3807704*x^56+12843980*x^55+37945904*x^54+99582920*x^53+234813592*x^52+502196500*x^51+981900168*x^50+1766948340*x^49+2943492972*x^48+4562339774*x^47+6609143792*x^46+8984070856*x^45+11500901864*x^44+13910043524*x^43+15941684776*x^42+17357937708*x^41+17999433372*x^40+17813777994*x^39+16859410792*x^38+15286065700*x^37+13299362332*x^36+11120136162*x^35+8948546308*x^34+6939692682*x^33+5193067630*x^32+3754272038*x^31+2625062144*x^30+1777171680*x^29+1166067584*x^28+742181216*x^27+458596528*x^26+275287664*x^25+160637648*x^24+91173896*x^23+50365440*x^22+27099984*x^21+14217528*x^20+7282760*x^19+3648664*x^18+1791312*x^17+863412*x^16+409202*x^15+190912*x^14+87752*x^13+39744*x^12+17724*x^11+7776*x^10+3356*x^9+1420*x^8+586*x^7+232*x^6+84*x^5+28*x^4+10*x^3+4*x^2+2*x+2) MPI(7,1,x) = (x^63+32*x^62+496*x^61+4976*x^60+36440*x^59+208336*x^58+971272*x^57+3807704*x^56+12843980*x^55+37945904*x^54+99582920*x^53+234813592*x^52+502196500*x^51+981900168*x^50+1766948340*x^49+2943492972*x^48+4562339774*x^47+6609143792*x^46+8984070856*x^45+11500901864*x^44+13910043524*x^43+15941684776*x^42+17357937708*x^41+17999433372*x^40+17813777994*x^39+16859410792*x^38+15286065700*x^37+13299362332*x^36+11120136162*x^35+8948546308*x^34+6939692682*x^33+5193067630*x^32+3754272038*x^31+2625062144*x^30+1777171680*x^29+1166067584*x^28+742181216*x^27+458596528*x^26+275287664*x^25+160637648*x^24+91173896*x^23+50365440*x^22+27099984*x^21+14217528*x^20+7282760*x^19+3648664*x^18+1791312*x^17+863412*x^16+409202*x^15+190912*x^14+87752*x^13+39744*x^12+17724*x^11+7776*x^10+3356*x^9+1420*x^8+586*x^7+232*x^6+84*x^5+28*x^4+10*x^3+4*x^2+2*x+2) allroots(a); allroots: only 3 out of 63 roots found. fpprec:40; fpprintprec:8; map('float,bfallroots(%i*a)); [x=0.161555*%i+0.411481, x=0.411481-0.161555*%i, x=0.424513-0.20753*%i, x=0.20753*%i+0.424513, x=0.357551*%i+0.467685, x=0.467685-0.357551*%i, x=0.456731-0.385631*%i, x=0.385631*%i+0.456731, x=0.6957*%i+0.358121, x=0.126497-0.691781*%i, x=0.686775*%i+0.37974, x=0.57103*%i+0.407889, x=0.411342-0.607333*%i, x=0.607333*%i+0.411342, x=0.691781*%i+0.126497, x=0.407889-0.57103*%i, x=0.37974-0.686775*%i, x=0.358121-0.6957*%i, x=0.00164296-0.822467*%i, x=-0.677577*%i-0.61351, x=0.845503*%i-0.239716, x=0.706577*%i-0.530524, x=-0.473795*%i-0.713626, x=0.473795*%i-0.713626 ,x=1.116797*%i-0.22244, x=0.822467*%i+0.00164296, x=-0.845503*%i-0.239716, x=-0.352383*%i-1.2950967, x=0.326845*%i-1.0300292, x=-0.706577*%i-0.530524, x=0.677577*%i-0.61351, x=1.0290328*%i-0.0124893, x=-1.0558428*%i-0.142756, x=1.1226818*%i-0.20742, x=-1.0290328*%i-0.0124893, x=1.006987*%i+0.00224111, x=1.0136286*%i-0.184951, x=-1.1049456*%i-0.157136, x=1.0558428*%i-0.142756, x=0.00224111-1.006987*%i, x=-1.0136286*%i-0.184951, x=-1.1226818*%i-0.20742, x=1.1049456*%i-0.157136, x=-0.326845*%i-1.0300292, x=-1.116797*%i-0.22244, x=0.102542*%i-1.3871109, x=0.352383*%i-1.2950967, x=-0.038694*%i-1.6292734, x=-0.102542*%i-1.3871109, x=0.038694*%i-1.6292734, x=-0.418759*%i-1.2385297, x=0.418759*%i-1.2385297, x=-0.0485021*%i-1.7677303, x=-0.435307*%i-1.2897288, x=0.435307*%i-1.2897288, x=0.0107183*%i-1.8616233, x=-0.0107183*%i-1.8616233, x=0.0485021*%i-1.7677303, x=9.22788967*10^-4*%i-1.9854688, x=-9.22788967*10^-4*%i-1.9854688, /* 3 real values */ x=1.63144623*10^-17*%i-1.9585096, /* = -1.9585096 */ x=-1.36578655*10^-17*%i-1.9205387, /* = -1.9205387 */ x=-4.53003421*10^-18*%i-1.9983914] /* = -1.9983914 */ ------------------------------------------------------- factor(MP(8,1,x)); factor(%); (x^9* (x+2)* (x^3+2*x^2+2*x+2)* (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* (x^15+8*x^14+28*x^13+60*x^12+94*x^11+116*x^10+114*x^9+94*x^8+70*x^7+48*x^6+32*x^5+20*x^4+10*x^3+4*x^2+2*x+2)*(x^31+16*x^30+120*x^29+568*x^28+1932*x^27+5096*x^26+10948*x^25+19788*x^24+30782*x^23+41944*x^22+50788*x^21+55308*x^20+54746*x^19+49700*x^18+41658*x^17+32398*x^16+23462*x^15+15872*x^14+10096*x^13+6096*x^12+3528*x^11+1976*x^10+1072*x^9+564*x^8+290*x^7+144*x^6+68*x^5+28*x^4+10*x^3+4*x^2+2*x+2)* (x^63+32*x^62+496*x^61+4976*x^60+36440*x^59+208336*x^58+971272*x^57+3807704*x^56+12843980*x^55+37945904*x^54+99582920*x^53+234813592*x^52+502196500*x^51+981900168*x^50+1766948340*x^49+2943492972*x^48+4562339774*x^47+6609143792*x^46+8984070856*x^45+11500901864*x^44+13910043524*x^43+15941684776*x^42+17357937708*x^41+17999433372*x^40+17813777994*x^39+16859410792*x^38+15286065700*x^37+13299362332*x^36+11120136162*x^35+8948546308*x^34+6939692682*x^33+5193067630*x^32+3754272038*x^31+2625062144*x^30+1777171680*x^29+1166067584*x^28+742181216*x^27+458596528*x^26+275287664*x^25+160637648*x^24+91173896*x^23+50365440*x^22+27099984*x^21+14217528*x^20+7282760*x^19+3648664*x^18+1791312*x^17+863412*x^16+409202*x^15+190912*x^14+87752*x^13+39744*x^12+17724*x^11+7776*x^10+3356*x^9+1420*x^8+586*x^7+232*x^6+84*x^5+28*x^4+10*x^3+4*x^2+2*x+2)* /* a = MPR(8,1,x) */ (x^127+64*x^126+2016*x^125+41696*x^124+637360*x^123+7685024*x^122+76185104*x^121+639097008*x^120+4634116312*x^119+29524775520*x^118+167453394320*x^117+854515874096*x^116+3958458557608*x^115+16771945556496*x^114+65418624260840*x^113+236221241425176*x^112+793548088258508*x^111+2490875091238112*x^110+7333879739219600*x^109+20324543852025936*x^108+53181959591958024*x^107+131760770157606224*x^106+309881648709683160*x^105+693434955498545848*x^104+1479594496462756340*x^103+3016191418506637264*x^102+5884917700519129288*x^101+11008161481780603512*x^100+19772322481956974532*x^99+34150590308701283528*x^98+56796799826096529620*x^97+91071943593142473900*x^96+140960183546144741182*x^95+210835921361505594848*x^94+305060580205223726864*x^93+427417353874088245520*x^92+580430565842543266504*x^91+764655844340519788496*x^90+978057923319151340728*x^89+1215604411161527170376*x^88+1469189341596552030212*x^87+1727958521630464742736*x^86+1979045408073272278264*x^85+2208653487832260558008*x^84+2403354418943890067404*x^83+2551425591643957182856*x^82+2644036970936308845148*x^81+2676118542978972739644*x^80+2646791812246207696810*x^79+2559320275988283588176*x^78+2420609646335251593480*x^77+2240353897304462193848*x^76+2029966595320794439668*x^75+1801452751402955781592*x^74+1566365198635995978988*x^73+1334957092752100409132*x^72+1115597581733327913554*x^71+914470757914434625800*x^70+735537050036491961156*x^69+580706779030058464252*x^68+450159936955994386738*x^67+342743629811082484420*x^66+256386228250001079082*x^65+188481251186354006062*x^64+136210493669590627494*x^63+96791719611591962624*x^62+67649247253332557888*x^61+46514944583399583872*x^60+31472438318100913024*x^59+20959276151880936672*x^58+13741246529613412192*x^57+8870996681175174400*x^56+5640327912934870240*x^55+3532711259263452288*x^54+2180053774542349632*x^53+1325752377025053872*x^52+794643419262469376*x^51+469540647020942704*x^50+273551723347748720*x^49+157160526459147600*x^48+89054477710037128*x^47+49778854852108736*x^46+27452220046644256*x^45+14938874048903424*x^44+8022838941878800*x^43+4252726080100416*x^42+2225318825517552*x^41+1149623838683192*x^40+586418795791480*x^39+295389697275984*x^38+146947468842816*x^37+72202154315704*x^36+35042955512616*x^35+16801812210008*x^34+7959065386896*x^33+3725380461620*x^32+1723231602514*x^31+787873523840*x^30+356124511056*x^29+159181600224*x^28+70382532248*x^27+30794621024*x^26+13338200216*x^25+5721586144*x^24+2431738892*x^23+1024406624*x^22+427893784*x^21+177269264*x^20+72854932*x^19+29707808*x^18+12019244*x^17+4824044*x^16+1920042*x^15+757456*x^14+296008*x^13+114520*x^12+43828*x^11+16568*x^10+6172*x^9+2252*x^8+794*x^7+264*x^6+84*x^5+28*x^4+10*x^3+4*x^2+2*x+2) fpprintprec:14; fpprec:80; map('float,bfallroots(%i*a)); x=0.231624835287*%i+0.443168987008, x=0.215759541627*%i+0.444791667649, x=0.384727430175-0.109283294449*%i, x=0.109283294449*%i+0.384727430175, x=0.346999967577*%i+0.465334074704, x=0.397391822297-0.133511204872*%i,x= 0.133511204872*%i+0.397391822297, x=0.444791667649-0.215759541627*%i, x=0.443168987008-0.231624835287*%i, x=0.465334074704-0.346999967577*%i, x=0.332463653001*%i+0.46035060144, x=0.46035060144-0.332463653001*%i, x=0.397744879285-0.561366453472*%i, x=0.395029433094*%i+0.464038063066, x=0.464038063066-0.395029433094*%i, x=0.403388665437*%i+0.454846689072, x=0.454846689072-0.403388665437*%i, x=0.527782719722*%i+0.372556286231, x=0.372556286231-0.527782719722*%i, x=1.054325792727*%i-0.175028727435, x=0.246040174088-0.621484277931*%i, x=0.652072647311*%i+0.14355621453, x=0.14355621453-0.652072647311*%i, x=0.621484277931*%i+0.246040174088, x=0.561366453472*%i+0.397744879285, x=0.702523240765*%i+0.358973540985, x=0.415941929341-0.618518985302*%i, x=0.618518985302*%i+0.415941929341, x=0.358973540985-0.702523240765*%i, x=0.609368848912*%i+0.419714756609, x=0.419714756609-0.609368848912*%i, x=0.381031131028-0.681357971068*%i, x=0.701723647143*%i+0.349822739013, x=0.668890639664*%i+0.383784424983, x=0.383784424983-0.668890639664*%i, x=0.349822739013-0.701723647143*%i, x=0.681357971068*%i+0.381031131028, x=-0.688133302278*%i-0.52331947714, x=-0.682380618944*%i-0.400339123112, x=0.0402543476872-0.819723496366*%i, x=0.458060975297*%i-0.670209187903, x=0.682380618944*%i-0.400339123112, x=1.1199208801277*%i-0.222876538518, x=0.60807426077*%i-0.624632770792, x=0.819723496366*%i+0.0402543476872, x=-0.458060975297*%i-0.670209187903, x=0.688133302278*%i-0.52331947714, x=-1.054325792727*%i-0.175028727435, x=-0.60807426077*%i-0.624632770792, x=-0.847206540295*%i-0.28852206209, x=0.857888656291*%i+0.0255481996683, x=-0.886155749107*%i-0.253701479219, x=0.0255481996683-0.857888656291*%i, x=0.199585983903*%i-1.2909367244877, x=0.847206540295*%i-0.28852206209, x=0.329673013642*%i-0.9184503692, x=-0.329673013642*%i-0.9184503692, x=0.684275632092*%i-0.629172036748, x=-0.694431267229*%i-0.605401481521, x=0.694431267229*%i-0.605401481521, x=-1.0952076237951*%i-0.19928665702, x=-0.684275632092*%i-0.629172036748, x=-0.982518322628*%i-0.018084985088, x=0.886155749107*%i-0.253701479219, x=1.0314503037037*%i-0.00789710379422, x=-1.0027556603635*%i-0.00136913381569, x=1.1047531613845*%i-0.197797414183, x=1.1202228182535*%i-0.206107193109, x=1.0952076237951*%i-0.19928665702, x=1.0027556603635*%i-0.00136913381569, x=1.0014697094519*%i-0.0676908885845, x=1.0357489203253*%i-0.0145572857228, x=-1.0314503037037*%i-0.00789710379422, x=1.1163504988128*%i-0.22666076429, x=-1.0357489203253*%i-0.0145572857228, x=0.982518322628*%i-0.018084985088, x=-1.0014697094519*%i-0.0676908885845, x=0.418037368955*%i-1.2914596913497, x=-1.0147729534904*%i-0.137775189807, x=0.37901588977*%i-1.0276641090312, x=1.0147729534904*%i-0.137775189807, x=-1.1047531613845*%i-0.197797414183, x=-1.1115934185722*%i-0.157145473737, x=-0.37901588977*%i-1.0276641090312, x=-0.347188794909*%i-1.149464372489, x=0.151354307079*%i-1.4192352121574, x=1.1054761969024*%i-0.151766572919, x=-1.1199208801277*%i-0.222876538518, x=-0.199585983903*%i-1.2909367244877, x=-1.1202228182535*%i-0.206107193109, x=1.1115934185722*%i-0.157145473737, x=-1.1054761969024*%i-0.151766572919, x=-1.1163504988128*%i-0.22666076429, x=0.0281753397792*%i-1.7400623825793, x=-0.0281753397792*%i-1.7400623825793, x=-0.0500368469414*%i-1.4912817508416, x=-0.0621023421633*%i-1.7702985346292, x=-0.0214315667527*%i-1.788503777584, x=-0.151354307079*%i-1.4192352121574, x=0.0500368469414*%i-1.4912817508416, x=0.0214315667527*%i-1.788503777584, x=0.347188794909*%i-1.149464372489, x=-0.418037368955*%i-1.2914596913497, x=-0.436655979636*%i-1.2808867803456, x=0.352619749858*%i-1.2769391594012, x=-0.440220125951*%i-1.2943253507119, x=0.0621023421633*%i-1.7702985346292, x=-0.352619749858*%i-1.2769391594012, x=0.409169938455*%i-1.2471511587439, x=0.440220125951*%i-1.2943253507119, x=-0.409169938455*%i-1.2471511587439, x=0.436655979636*%i-1.2808867803456, x=0.00109874682389*%i-1.9668027797244, x=-0.00334194434366*%i-1.9073768267543, x=-0.00109874682389*%i-1.9668027797244, x=0.00334194434366*%i-1.9073768267543, x=-1.13712571273607*10^-4*%i-1.9963788258049, x=1.13712571273607*10^-4*%i-1.9963788258049, /* 9 real values = on main antenna */ x=-2.59942168948555*10^-36*%i-1.6094791752545, = -1.6094791752545 x=1.80255970181048*10^-35*%i-1.6443184654986, = -1.6443184654986 x=-1.96547114300395*10^-31*%i-1.8447687460482, = -1.8447687460482 x=7.17480153826594*10^-31*%i-1.875762032976, = -1.875762032976 x=-9.81569768441473*10^-30*%i-1.9342580450854, = -1.9342580450854 x=1.6361593078493*10^-29*%i-1.9478631515087, = -1.9478631515087 x=-3.05606977973738*10^-29*%i-1.9802055873813, = -1.9802055873813 x=3.37822421557851*10^-29*%i-1.9898637721811, = -1.9898637721811 x=-1.02888234543023*10^-29*%i-1.9995982370157 = -1.9995982370157 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ----------------------------------------------------------- %%%%%%%%%%%%%%%%%%%% period 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ----------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% factor(MP(2,2,x)) = x^3* /* center of period 1 */ (x+1)^2* /* center of period 2 */ (x+2)* /* MPI(2,1,x) */ (x^2+1) /* MPI(2,2,x) */ M(2,2); [x=0.0,x=0.0,x=0.0,x=1.0*%i-3.0212977014291249*10^-17,x=-1.0*%i-3.0212977014291249*10^-17,x=-0.99999997019768,x=-1.000000029802323,x=-1.999999999999999] solve(x^2+1,x) = [x=-%i,x=%i] ------------------------------------------------------------------ M(3,2) = [x=0.6062907292072*%i+0.41964337760708, x=-1.839286755214161, x=0.41964337760708-0.6062907292072*%i, x=-2.0, x=-1.0*%i, x=%i, x=-1.115142508039935*%i-0.22815549365396, x=1.115142508039935*%i-0.22815549365396, x=-1.543689012692072, x=-1.0, x=0.0] MP(3,2,x) = x^6 * MPI(3,2)* MPI(3,1) * ..... (((x^2+x)^2+x)^2+x)^2-(x^2+x)^2 = x^4 * /* center of period 1 */ (x+1)^3 * /* center of period 2 */ (x+2) * /* MPI(2,1,x) */ (x^2+1) * /* MPI(2,2,x) */ ( x^3+x^2-x+1) * /* MPI(3,2,x) */ (x^3+2*x^2+2*x+2) /* MPI(3,1,x) */ allroots(x^3+2*x^2+2*x+2); it is M(3,1) [x=1.115142508039937*%i-0.22815549365396, x=-1.115142508039937*%i-0.22815549365396, x=-1.543689012692076] allroots( x^3+x^2-x+1); it is M(3,2) [x=0.6062907292072*%i+0.41964337760708, x=0.41964337760708-0.6062907292072*%i, x=-1.839286755214161] so MPI(3,2,x) = x^3+x^2-x+1 ----------------------------------------------- factor(MP(4,2,x)) = x^5 * /* center of period 1 */ (x+1)^3* /* center of period 2 */ (x+2)* /* MPI(2,1,x) */ (x^2+1)* /* MPI(2,2,x) */ (x^3+x^2-x+1)* /* MPI(3,2,x) */ (x^3+2*x^2+2*x+2)* /* MPI(3,1,x) */ (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* /* MPI(4,1,x) */ (x^8+4*x^7+6*x^6+6*x^5+4*x^4+1) /* MPI(4,2,x) */ [x=0.34658833134003*%i+0.4704260283452, x=0.4704260283452-0.34658833134003*%i, x=0.68399196808937*%i-0.50629445819589, x=-0.68399196808937*%i-0.50629445819589, x=1.103446239366782*%i-0.15248775721021, x=-1.103446239366782*%i-0.15248775721021, x=-1.661239227236016, x=-1.962048398642176] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% period 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% factor(MP(4,3,x)); x^5* /* center of period 1 */ (x+2)* /* MPI(2,1,x) */ (x^3+2*x^2+x+1)^3* /* MPI(3,2,x) */ (x^3+2*x^2+2*x+2)* /* MPI(3,1,x) */ (x^6+2*x^5+2*x^4+2*x^3+x^2+1)* /* ???? */ (x^7+4*x^6+6*x^5+6*x^4+6*x^3+4*x^2+2*x+2)* /* MPI(4,1,x) */ (x^12+6*x^11+14*x^10+18*x^9+18*x^8+16*x^7+10*x^6+6*x^5+5*x^4+2*x^3+1)* /* ???? */ (x^21+10*x^20+45*x^19+125*x^18+249*x^17+384*x^16+472*x^15+473*x^14+ 400*x^13+295*x^12+201*x^11+135*x^10+84*x^9+44*x^8+19*x^7+8*x^6+6*x^5+5*x^4+2*x^3-x^2-x+1) /* MPI(4,3,x) */ map('float,bfallroots(%i*P)); x=0.19747355193629*%i+0.43525070045515, x=0.43525070045515-0.19747355193629*%i, x=0.46561838717476-0.39445372332124*%i, x=0.39445372332124*%i+0.46561838717476, x=0.62033603933226*%i+0.41838821346836, x=0.41838821346836-0.62033603933226*%i, x=-0.69337549987481*%i-0.60329263492103, x=-0.99545453492254*%i-0.027219872189945, x=0.69337549987476*%i-0.60329263492102, x=0.37244732766414*%i-1.010330521377608, x=-1.06275228084941*%i-0.17300671609218, x=0.99545453492235*%i-0.027219872190274, x=1.062752280849784*%i-0.17300671609074, x=-0.37244732766285*%i-1.010330521376742, x=-1.122483745619165*%i-0.20638467176339, x=1.122483745619027*%i-0.20638467176435, x=-0.1256954358636*%i-1.419116186447585, x=0.12569543586479*%i-1.419116186453749, /* 3 real values on main antennae */ x=-8.192320187328848*10^-12*%i-1.790327492049422, = -1.790327492049422 x=8.5832551076749987*10^-11*%i-1.981056902605377, = -1.981056902605377 x=-8.0131897484693474*10^-11*%i-1.988429001953127 = -1.988429001953127 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% ???? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P:x^6+2*x^5+2*x^4+2*x^3+x^2+1; map('float,bfallroots(%i*P)); [x=0.555624571006*%i+0.39501405207669, x=0.39501405207669-0.555624571006*%i, x=-1.112217114596038*%i-0.15578849668713, x=1.112217114596038*%i-0.15578849668713, x=0.412602181602*%i-1.23922555538956, x=-0.412602181602*%i-1.239225555389575] P:x^12+6*x^11+14*x^10+18*x^9+18*x^8+16*x^7+10*x^6+6*x^5+5*x^4+2*x^3+1; map('float,bfallroots(%i*P)); [x=0.32495732375097*%i+0.45481578189569, x=0.60525138127893*%i-0.60703102261609, x=0.45481578189569-0.32495732375097*%i, x=0.38183759567086-0.68613603770228*%i, x=0.68613603770228*%i+0.38183759567086, x=-0.60525138127893*%i-0.60703102261609, x=-1.030984699759986*%i-0.0075674155100787, x=1.030984699759986*%i-0.0075674155100787, x=0.34772469142135*%i-1.283657575168204, x=-0.34772469142135*%i-1.283657575168204, x=-3.5428992773930216*10^-15*%i-1.924661063186362, x=2.9877877650804433*10^-15*%i-1.952133665357998] */