ireducible polynomials with coefficients in ${0,-1}$












4














I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question
























  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28
















4














I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question
























  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28














4












4








4







I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question















I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio




dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 26 at 9:41

























asked Feb 10 at 6:08









Linas

417210




417210












  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28


















  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28
















In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
– Andy Walls
Feb 10 at 12:02






In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
– Andy Walls
Feb 10 at 12:02














Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
– Andy Walls
Feb 10 at 12:37




Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
– Andy Walls
Feb 10 at 12:37












Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
– Linas
Feb 11 at 8:28




Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
– Linas
Feb 11 at 8:28










1 Answer
1






active

oldest

votes


















0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04











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1 Answer
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1 Answer
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0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04
















0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04














0












0








0






Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer












Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)






share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Feb 12 at 1:04









Andy Walls

1,539128




1,539128












  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04


















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04
















The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
– Linas
Feb 12 at 20:01




The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
– Linas
Feb 12 at 20:01












Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
– Linas
Feb 12 at 20:04




Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
– Linas
Feb 12 at 20:04


















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Human spaceflight

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Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e...
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Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

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0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
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張江高科駅

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張江高科駅 ちょうこうこうか Zhangjiang Hi-Tech Park 所在地 中国上海市浦東新区 所属事業者 上海地鉄運営有限公司（上海軌道交通・駅詳細） 上海浦東現代有軌交通有限公司（駅詳細） テンプレートを表示 張江高科駅 各種表記 繁体字： 張江高科站 簡体字： 张江高科站 拼音 ： Zhàngjiāng Gāokē Zhàn 発音： ヂャンジャン ガオクー ヂャン 英文： Zhangjiang Hi-Tech Park Station テンプレートを表示 張江高科駅 （ちょうこうこうか-えき）は中華人民共和国上海市浦東新区に位置する、上海軌道交通2号線の駅である。 なお、張江高科駅前にある張江軌道電車の 張江地鉄駅駅 についてもここで述べる。 目次 1 利用可能な鉄道路線 2 沿革 3 駅構造 3.1 上海軌道交通 3.2 張江軌道電車 4 駅周辺 5 隣の駅 6 注釈 7 関連項目 利用可能な鉄道路線 上海軌道交通 ■ 2号線 上海浦東現代有軌交通有限公司 ■ 張江軌道電車 沿革 高架駅時代の姿 2000年12月27日 - 2号線の駅が高架駅として開業。 2010年 1月1日 - 張江軌道電車の開業に伴い、張江地鉄駅駅（停留所）を設置。 2月13日 - 駅地下化に伴い高架駅の運用停止。 2月24日 - 10日間の閉鎖を経て地下新駅の開業。 2017年2月9日 - リニューアル工事を経て再開業。乗降客の増加に対応し、ホームへのエレベーター、エスカレーターを増設、券売機エリアを拡張した [1] 。 駅構造 上海軌道交通 上海軌道交通 張江高科駅 5号口 ちょうこうこうか Zhangjiang Hi-Tech Park ◄ 竜陽路 (2.60km) (1.61km) 金科路 ► 所在地 中国上海市浦東新区松涛路 祖冲之路 （中国語版） 北緯31度12分11秒 東経121度34分50秒  /  北緯31.2...
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