# Revision history [back]

You can define the ring of (numerical) polynomials over C as follows:

sage: R.<x> = CDF[]
sage: R
Univariate Polynomial Ring in x over Complex Double Field


You can construct a random polynomial of degree 30 as follows:

sage: P = R.random_element(degree=30)
sage: P
(0.7368648392813568 + 0.7644659236040987*I)*x^30 + (-0.9521231496916671 - 0.5668264771579159*I)*x^29 + (-0.5602325090820397 + 0.08373185660033844*I)*x^28 + (0.11778992480095218 - 0.2031034081348333*I)*x^27 + (0.3516830229409873 + 0.2968075058912192*I)*x^26 + (-0.7794485386929075 + 0.35282724454074765*I)*x^25 + (0.9529334155548153 - 0.10858391867189021*I)*x^24 + (0.4346153404358597 - 0.7919849906353298*I)*x^23 + (-0.4122547991033889 - 0.09369196690243964*I)*x^22 + (0.9072745543379173 - 0.93084223388219*I)*x^21 + (0.0917339388366103 + 0.8400827723548798*I)*x^20 + (0.620441590530908 + 0.5497959418695815*I)*x^19 + (0.08961053619498971 - 0.3245694231244378*I)*x^18 + (-0.1573830771827882 - 0.0999821789852311*I)*x^17 + (-0.42868409209771885 + 0.17826524252278442*I)*x^16 + (0.08423322767931607 - 0.7883353014539329*I)*x^15 + (-0.4854333236724999 + 0.14977131931558807*I)*x^14 + (-0.44684046421460066 - 0.46310410665013624*I)*x^13 + (0.6526594274639133 - 0.4196336531832292*I)*x^12 + (-0.27787355156700233 + 0.10848474232286232*I)*x^11 + (-0.11101383721798008 - 0.5037475397998767*I)*x^10 + (-0.13186950326025015 - 0.4838574269429168*I)*x^9 + (0.8187926299228285 + 0.8711795580308974*I)*x^8 + (-0.6860149090655201 - 0.9849078620505949*I)*x^7 + (-0.6918369434398146 - 0.9814493553853534*I)*x^6 + (0.3499220158155265 + 0.5747414242705378*I)*x^5 + (-0.893601061756806 + 0.5607846129706036*I)*x^4 + (-0.12889260181776785 + 0.7202322287337615*I)*x^3 + (-0.6940468382565192 - 0.5557200414145016*I)*x^2 + (-0.08633744827278456 + 0.7424063585043299*I)*x + 0.6230123145504038 + 0.6286232382291024*I


You can find its roots as follows:

sage: P.roots(multiplicities=False)
[-1.0120138172468134 + 0.20426669296859173*I,
-0.962581077360827 - 0.5296864154447034*I,
-0.9364728333831983 - 0.24073599552387334*I,
-0.9201821425300504 + 0.002487918823541706*I,
-0.8599059855418224 + 0.492490723674138*I,
-0.8215080191130956 - 0.6427711927416188*I,
-0.7327655570630038 + 0.6808285540382258*I,
-0.7017713764174295 + 0.9067735719613969*I,
-0.6414579210848824 + 0.1305690899208492*I,
-0.5346277142391518 - 0.7813046856018692*I,
-0.36837899733455526 - 0.896830748646684*I,
-0.3534478170558439 + 0.7077101949275129*I,
-0.28232967917268137 + 0.9677582197606973*I,
-0.22178572802727103 - 0.9734268214408631*I,
0.015053958533473745 + 1.035138545876896*I,
0.09170010634309536 - 1.0719575404124397*I,
0.23422262713071448 + 0.9812411153412857*I,
0.243546812209695 - 1.029432097206379*I,
0.39934906552852045 - 0.8111814833649242*I,
0.4716233302309813 + 0.8546354765171169*I,
0.582436456639921 - 0.7624604937976489*I,
0.6491856986029882 + 0.6138112252444622*I,
0.7124110338205422 + 0.4351851664034465*I,
0.7570474447707015 + 0.7787758630291586*I,
0.831513193382847 - 0.6444495428445836*I,
0.9648971684027969 - 0.38704090145535897*I,
0.9660222981279798 + 0.24237600403657947*I,
0.9892771693661305 + 0.06311132544291641*I,
1.1525531039685055 + 0.017246602835138347*I,
1.2950663367376647 - 0.618272306837997*I]


You can construct the list of its absolute values as follows:

sage: [abs(r) for r in P.roots(multiplicities=False)]
[1.0324228049373916,
1.0986946933519746,
0.9669204658117574,
0.9201855058471996,
0.9909517732340606,
1.0430868763845154,
1.0002364128602976,
1.1466130016559397,
0.6546117565132008,
0.946712102265528,
0.9695403433562648,
0.7910619946539459,
1.0081002031822746,
0.9983730193955808,
1.0352480045031056,
1.0758726123247224,
1.0088083888906871,
1.0578494658958941,
0.9041543425171509,
0.9761303003901686,
0.9594676808655002,
0.8934239147833227,
0.8348147160698889,
1.0860997553036549,
1.052012074095318,
1.0396283975493898,
0.995964461119574,
0.9912882311559164,
1.152682134318386,
1.4350809949107028]

 2 No.2 Revision slelievre 14559 ●16 ●136 ●287 http://carva.org/samue...

## Polynomials with complex coefficients

You can define the ring of (numerical) polynomials over C as follows:

sage: R.<x> = CDF[]
sage: R
Univariate Polynomial Ring in x over Complex Double Field


You can construct a random polynomial of degree 30 as follows:

sage: P = R.random_element(degree=30)
sage: P
(0.7368648392813568 + 0.7644659236040987*I)*x^30 + (-0.9521231496916671 - 0.5668264771579159*I)*x^29 + (-0.5602325090820397 + 0.08373185660033844*I)*x^28 + (0.11778992480095218 - 0.2031034081348333*I)*x^27 + (0.3516830229409873 + 0.2968075058912192*I)*x^26 + (-0.7794485386929075 + 0.35282724454074765*I)*x^25 + (0.9529334155548153 - 0.10858391867189021*I)*x^24 + (0.4346153404358597 - 0.7919849906353298*I)*x^23 + (-0.4122547991033889 - 0.09369196690243964*I)*x^22 + (0.9072745543379173 - 0.93084223388219*I)*x^21 + (0.0917339388366103 + 0.8400827723548798*I)*x^20 + (0.620441590530908 + 0.5497959418695815*I)*x^19 + (0.08961053619498971 - 0.3245694231244378*I)*x^18 + (-0.1573830771827882 - 0.0999821789852311*I)*x^17 + (-0.42868409209771885 + 0.17826524252278442*I)*x^16 + (0.08423322767931607 - 0.7883353014539329*I)*x^15 + (-0.4854333236724999 + 0.14977131931558807*I)*x^14 + (-0.44684046421460066 - 0.46310410665013624*I)*x^13 + (0.6526594274639133 - 0.4196336531832292*I)*x^12 + (-0.27787355156700233 + 0.10848474232286232*I)*x^11 + (-0.11101383721798008 - 0.5037475397998767*I)*x^10 + (-0.13186950326025015 - 0.4838574269429168*I)*x^9 + (0.8187926299228285 + 0.8711795580308974*I)*x^8 + (-0.6860149090655201 - 0.9849078620505949*I)*x^7 + (-0.6918369434398146 - 0.9814493553853534*I)*x^6 + (0.3499220158155265 + 0.5747414242705378*I)*x^5 + (-0.893601061756806 + 0.5607846129706036*I)*x^4 + (-0.12889260181776785 + 0.7202322287337615*I)*x^3 + (-0.6940468382565192 - 0.5557200414145016*I)*x^2 + (-0.08633744827278456 + 0.7424063585043299*I)*x + 0.6230123145504038 + 0.6286232382291024*I


You can find its roots as follows:

sage: P.roots(multiplicities=False)
[-1.0120138172468134 + 0.20426669296859173*I,
-0.962581077360827 - 0.5296864154447034*I,
-0.9364728333831983 - 0.24073599552387334*I,
-0.9201821425300504 + 0.002487918823541706*I,
-0.8599059855418224 + 0.492490723674138*I,
-0.8215080191130956 - 0.6427711927416188*I,
-0.7327655570630038 + 0.6808285540382258*I,
-0.7017713764174295 + 0.9067735719613969*I,
-0.6414579210848824 + 0.1305690899208492*I,
-0.5346277142391518 - 0.7813046856018692*I,
-0.36837899733455526 - 0.896830748646684*I,
-0.3534478170558439 + 0.7077101949275129*I,
-0.28232967917268137 + 0.9677582197606973*I,
-0.22178572802727103 - 0.9734268214408631*I,
0.015053958533473745 + 1.035138545876896*I,
0.09170010634309536 - 1.0719575404124397*I,
0.23422262713071448 + 0.9812411153412857*I,
0.243546812209695 - 1.029432097206379*I,
0.39934906552852045 - 0.8111814833649242*I,
0.4716233302309813 + 0.8546354765171169*I,
0.582436456639921 - 0.7624604937976489*I,
0.6491856986029882 + 0.6138112252444622*I,
0.7124110338205422 + 0.4351851664034465*I,
0.7570474447707015 + 0.7787758630291586*I,
0.831513193382847 - 0.6444495428445836*I,
0.9648971684027969 - 0.38704090145535897*I,
0.9660222981279798 + 0.24237600403657947*I,
0.9892771693661305 + 0.06311132544291641*I,
1.1525531039685055 + 0.017246602835138347*I,
1.2950663367376647 - 0.618272306837997*I]


You can construct the list of its absolute values as follows:

sage: [abs(r) for r in P.roots(multiplicities=False)]
[1.0324228049373916,
1.0986946933519746,
0.9669204658117574,
0.9201855058471996,
0.9909517732340606,
1.0430868763845154,
1.0002364128602976,
1.1466130016559397,
0.6546117565132008,
0.946712102265528,
0.9695403433562648,
0.7910619946539459,
1.0081002031822746,
0.9983730193955808,
1.0352480045031056,
1.0758726123247224,
1.0088083888906871,
1.0578494658958941,
0.9041543425171509,
0.9761303003901686,
0.9594676808655002,
0.8934239147833227,
0.8348147160698889,
1.0860997553036549,
1.052012074095318,
1.0396283975493898,
0.995964461119574,
0.9912882311559164,
1.152682134318386,
1.4350809949107028]


## Monic integer polynomials

Likewise define the ring of polynomials in x with integer coefficients:

sage: Zx = ZZ['x']
sage: x = Zx.gen()


Pick a random monic polynomial of degree 56:

sage: q = Zx.random_element(degree=56)  # not necessarily monic
sage: p = (1 - q[56])*x^56 + q  # now monic by making leading coefficient 1
x^56 - x^55 + x^54 - 7*x^53 - 2*x^52 - x^51 + 5*x^50 - x^49 - x^48 + x^47
- 3*x^44 + 3*x^43 + 3*x^42 + 20*x^41 - x^40 + x^39 + 125*x^38 + 9*x^37
- x^36 - 2*x^35 + 3*x^34 + 10*x^32 + x^30 + 99*x^29 + 7*x^27 + 13*x^25
+ x^23 + x^22 + 2*x^21 + 3*x^20 + 2*x^19 + x^17 + 2*x^14 - x^12 + 5*x^11
+ x^10 - x^8 + 7*x^7 - x^6 + x^5 - 5*x^4 - x^3 + 5*x^2 + x - 1


Get the sorted list of absolute values of its complex roots:

sage: sorted(set(abs(z) for z in p.complex_roots()))
[0.392775879097575, 0.618313001041181, 0.829872478969433,
0.876156325725046, 0.877139969220623, 0.896140603140608,
0.897355058131500, 0.898822808765931, 0.902218110575174,
0.908268048474002, 0.909765827992206, 0.914374431830909,
0.923321003752644, 0.935403911031256, 0.938567671569716,
0.970873370350356, 0.972725264654593, 0.975123271228316,
0.977132068373407, 1.01049931640877, 1.15177593823796,
1.16627154233445, 1.18058797182489, 1.21320658003016,
1.21544302417961, 1.23985793698233, 1.26758074663977,
1.28869109993426, 1.80658377491657, 2.17180464952677]