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3 votes

Equation $wxyz(w+x+y+z)=1$ in $\mathbb{Q}_+^4$

Let \begin{align*} w &= \frac{c_1t^{t_1}}{ABC}\\ x &= \frac{c_2B^2t^{t_2}}{A}\\ y &= \frac{c_3A^2t^{t_3}}{C}\\ z &= \frac{c_4Ct^{t_4}}{B} \end{align*} Then $$wxyz(w+x+y+z) = \frac{c_1t^...
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  • 1,019
4 votes

Equation $wxyz(w+x+y+z)=1$ in $\mathbb{Q}_+^4$

I finally found the following parametrization : $$w=\frac{32916734851\,t^4}{(219896613895728\,t^5+1)(370352191824384\,t^5+1)(479774430317952\,t^5+1)}$$ $$x=\frac{32\,(219896613895728\,t^5+1)(...
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  • 523
6 votes

Closed form roots for polynomial $x^9 + ax^6 + bx^5 + cx^3 + d = 0$

There is no way to transform your first polynomial to the special shape of the second polynomial while preserving its Galois group. The Galois group of the second polynomial is solvable, but for ...
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4 votes

Stothers-Mason theorem

Look up Belyi maps. See, for instance, the third page of Granville and Tucker’s survey paper on the $abc$ conjecture: It's as easy as $abc$.
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  • 42.5k
12 votes
Accepted

Resultant of linear combinations of Chebyshev polynomials of the second kind

Since $U_n + t U_{n-1}$ is of degree $n$ and $\sum_{k=0}^{n-1} U_k$ is of degree $n-1$ with leading coefficient $2^{n-1}$, the resultant factors as $$ 2^{n(n-1)} (-1)^{n(n-1)} \prod_{j=1}^{n-1} (U_n(...
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  • 91.6k
2 votes

Explicit and fast error bounds for polynomial approximation

After analyzing the proof of Güntürk and Li (2021), Theorem 3.3, I believe I found explicit error bounds for the Micchelli–Felbecker polynomials (iterated Bernstein polynomials) when $f(x)$ has a ...
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  • 603
2 votes
Accepted

On well separated circular regions in the Riemann sphere and complex polynomials

Let $\{ z_1,\ldots,z_n\}$ be any finite set, and $$L_k(z)=\prod_{j\neq k}(z-z_j).$$ Then $L_k,\; 1\leq k\leq n$ are linearly independent since the set of their linear combinations consists of all ...
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1 vote
Accepted

Determinant of a certain Vandermonde matrix

In Appendix B of https://arxiv.org/abs/2103.10776 (J. Phys. A: Math. Theor. 54, 375201 (2021)), I derived a transformation of the block Vandermonde determinant above to a Hankel matrix, which in my ...
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1 vote

Is it possible to solve for y in this equation?

Yes, it is. In fact, this is a trinomial equation. A closed form expression can be obtained using confluent Fox-Wright Function $\ _1\Psi_1^*(\zeta)$. See here A linearly convergent series can be ...
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  • 1,465
16 votes
Accepted

A statement on complex polynomials

The conjecture is easily seen to be true for $n<3$. We give a counterexample for $n=3$. Let $p_i = z^2 - \omega_i$ where the $\omega_i$ are the cube roots of unity. These are linearly dependent ...
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2 votes

A question about generalized harmonic numbers modulo $p$

Here is what I've learned: \begin{align*} 2\sum_{n = 0}^{\infty} B_{2n+1}\left(\frac{1}{3}\right) \frac{x^{2n+1}}{(2n+1)!} & = \frac{xe^{\frac{1}{3}x}}{e^x -1} + \frac{xe^{-\frac{1}{3}x}}{e^{-x} -...
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