55
votes
Accepted
Minimal polynomial of cos(π/n)
The minimal polynomial of $\cos(2\pi/n)$ (by William Watkins and Joel Zeitlin, The American Mathematical Monthly
Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take ...
- 16.5k
33
votes
Accepted
Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?
There is no such solution. Let
$$
Q(a,b,c,d) = 2(ad-bc)(bd+ac) + (a^2-b^2)(c^2-d^2)
$$
be the difference between the two sides of the equation,
so we seek to solve $Q(a,b,c,d) = 0$. This is a ...
- 77.5k
17
votes
Minimal polynomial of cos(π/n)
I guess you mean a polynomial $p(x)$ with rational coefficients. Then, once $\cos {\pi/n}$ is a root of $p(x)$, $\deg p=d$, $e^{i\pi/n}$ is a root of a polynomial $t^dp((t+1/t)/2)$. But $e^{i\pi/n}$ ...
- 103k
5
votes
Minimal polynomial of cos(π/n)
There is a Wikipedia entry dedicated to this, which contains an alternative method to compute the minimal polynomial of $2\cos(\pi/n)$, which is essentially the same as for $\cos(π/n)$. In fact, ...
- 13.2k
4
votes
speeding up Gosper and WZ algorithms
The group at RISC has been working aggressively towards developing improved algorithms for many computer algebra problems, including the Zeilberger's. So, it is a good place to ask.
Meantime, I just ...
- 41.8k
4
votes
Accepted
Fast Symbolic Linear Algebra CAS?
I found out that the best solution is actually SymEngine.jl in Julia. Using Julia as the driver ends up increasing the speed and reducing the memory requirement, making it more efficient than using ...
- 424
4
votes
Accepted
Symbolic powers of a prime ideal of height one
Take $E$ an elliptic curve $zy^2 - x(x-z)(x-tz)$ say over $\mathbb{C}$ and choose a point $Q$ of infinite order (or so for instance the divisor $Q - O$ has infinite order in the divisor class group, ...
- 20.2k
4
votes
Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
I'm not sure if this is exactly what you're looking for, but my go-to volume for these kinds of question is Symbolic Integration I by Manuel Bronstein. Risch's original treatment is sketchy in many ...
- 78.6k
4
votes
Accepted
Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$
First, a few simplifications. Note that $g(-t)\ge g(t)$ and $|-t-1|\ge|t-1|$ if $t\ge0$. So, without loss of generality (wlog) $t\ge0$ and
$$g(t)=\sqrt{(t-1)^2 + a^2} + bt. \tag{1}\label{1}$$
Since $g(...
- 118k
3
votes
Accepted
Is the smallest root of this quartic always the closest point on the Hyperbola?
No.
E.g., take $a=4$ and $b=27/8$. Then the positive roots of $g$ are $x_1\approx0.338$, $x_2\approx0.826$, $x_3\approx3.78$, and $f(x_1)\approx13.6$ and $f(x_3)\approx9.72$, so that $x_1$ is not a ...
- 118k
3
votes
2
votes
Accepted
Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals
Just expanding my comments to this question and the previous one:
I assume that your polynomials have rational coefficients (which seem to be the case, since you mention they are floating point ...
- 20.5k
2
votes
Accepted
CAS implementing free algebras with involution
I've figured how to do this with MAGMA. I'm going to document it, in case anyone has the same problem in the future:
1) Define a parameter field:
R$<$a1,...,an$>$:=FieldOfFractions(...
- 2,962
2
votes
Accepted
Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$
We have
$$\alpha = \left\{\begin{array}{ll}
\sqrt{a^2 + b^2} & \mathrm{if} ~ (a^2 + b^2)R^2 \ge a^2 + cb^2, \\[6pt]
R\sqrt{a^2 + b^2/c}& \mathrm{if} ~ a^2c^2 + b^2c \ge (a^2c^2 + b^2)...
- 848
1
vote
Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
J. H. Davenport, On the integration of algebraic functions. Lecture Notes in Computer Science, 102. Springer-Verlag, Berlin-New York, 1981.
J. H. Davenport, Integration in closed form. Computers in ...
- 88.9k
1
vote
Minimal polynomial of cos(π/n)
In PARI/ GP :
r2n = Mod(x, polcyclo(2*n))
minpoly((r2n + 1/r2n) / 2)
- 2,768
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