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### how to solve this multivariate quadratic equation?

This has been posted on math.stackexchange but got just one partial(insightful though) comment. I'm posting it here in a hope of getting further ideas and comments: The problem was: Any hope to ...
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### How to handle a polynomial whose roots exhibit obvious symmetry

I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one All others have their roots arranged in a similar trident-...
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### System of two variables quadratic equations

Let $\mathcal{P}_{2,Z}$ be the set of all 2 variables quadratic equations $P(x,y)$ with integral coefficients: $$P(x,y)=a_1x^2+a_2y^2+a_3xy+a_4x+a_5y+a_6\ \ \ \ \ \ (a_i\in \mathbb{Z})$$ Consider a ...
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I have a question about the solution of Pell-equation over a prime field. I want prove that the matrix $M$ is of rank $\frac{p-1}{2}$, with $M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} \right)^{(... 0answers 210 views ### An equation about generating functions and subfactorial As I promised, I clone the problem from Math.SE to here, in order to find a solution. Suppose$G_n(w)$is a formal power series (really a probability generating function, see the following ... 0answers 130 views ### satisfiable polynomial equations for given free coefficients Let$F$be a finite field,$n, k, m$be natural numbers. I give you$m$vectors$c^{(1)},\ldots,c^{(m)}\in F^n$. I ask for polynomials$p_1,\ldots,p_n$on$k$variables over$F$such that the system ... 0answers 31 views ### Does this system have a closed-form solution?$x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $I am interested in solving the following system of$n$equations: $$x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha$$ for all$j\in\{1,\dots,n\}$, where$n$is a positive integer,$0<\alpha<...
Is it possible to solve a function with both exponential and logarithm such as $a x^2 - b.\log(x) = c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?