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12
votes
3answers
480 views

Existence of solutions of a polynomial system

Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional real vector, and define $$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \...
11
votes
3answers
1k views

Can Fuchsian functions solve the general equation of degree n?

In the classic textbook Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that “the algebraic solution of the general equation of degree n is impossible if n is ...
8
votes
2answers
526 views

When is the degree of this number 3?

I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that ...
7
votes
1answer
713 views

Can roots of any polynomial be expressed using Eulerian function?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it is claimed that roots of any ...
7
votes
2answers
369 views

Bivariate polynomials with special properties

I recently came across some polynomials with some remarkable properties. A polynomial $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if the set of solutions to the system $P(u,v)=P(v,u)=0$...
6
votes
1answer
802 views

Is there an analytical method of solving general square root equations?

Equations such as $\sqrt{x+1}+\sqrt{x+2}=x+3$ are easily solvable by squaring both sides. But if we increase an extra square root, like if trying to solve $\sqrt{x+1}+\sqrt{x+2}+\sqrt{x+3}=x+4$ we ...
4
votes
0answers
1k views

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 ...
3
votes
3answers
514 views

For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer. Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
3
votes
2answers
160 views

Asymptotics of solution of transcendental equation [closed]

It seems the real solution of the following equation $$t=u^2+u\log(u) $$ has no closed form in view of the output of Reduce[t == u^2 + u*Log[u], u, Reals] The ...
3
votes
1answer
159 views

Existence of solution for this set of polynomial equations

We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where $$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$ I'm interested in proving that a solution for ...
3
votes
0answers
192 views

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-...
2
votes
3answers
599 views

Solving a system of algebraic equations

I am trying to find complex solutions with positive real part $\{t_j \;|\;{\rm Re}\;t_j>0, j = 1, 2, 3, \dots, n\}$ of the system of equations $$0 = 1 + \sum_j \left(t_j^{2l+1} + {t_j^*}^{2l+1}\...
2
votes
1answer
329 views

Equations of the Hirzebruch surface embedded in a large space.

Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$ and let $D$ be the very ample divisor $3C_0+5f$ on $\mathbb{F}_1$ (notation as in [Hartshorne, Algebraic ...
2
votes
1answer
109 views

Roots of modified polynomials

Consider the following two polynomials: $$ g=x^3 - x^2 - (c + 2)x + c $$ and $$ h=x^3 - x^2 - cx + c $$ The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...
2
votes
1answer
156 views

Nilradical and Newton's identities

Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...
2
votes
0answers
427 views

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 ...
1
vote
2answers
3k views

Real root of a cubic equation [closed]

I have a function f(x,n) can be expressed as a cubic function of x with coefficients that are functions of n. For example x^3 + (n-2)x^2 + (3n-6)x + n. I want to prove that for every positive value ...
1
vote
0answers
30 views

Solving exponential system [closed]

How can i solve the following system? $ \begin{cases} x^y = 16 \\ \frac{x}{y}=2\\ \end{cases}$ I tried everything, nothing works.
1
vote
0answers
46 views

About the rank of a Pell equation-related matrix

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)^{(...
1
vote
0answers
85 views

How to solve a system of quadratic equations with multiple unknowns? [closed]

While solving a problem i have ended up with these 4 equations: a^2+b^2= p; (c-q)^2+d^2=r; (d-b)/(c-a)=s; (c-a)^2-(d-b)^2=t Here a,b,c,d are unknowns . The rest (p,q,r,s,t) are known values. I am ...
1
vote
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 ...
1
vote
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 ...
0
votes
3answers
603 views

Is there any method to solve a Bivariate Cubic Equation System? [closed]

$f(x, y) = 0$ and $g(x, y) = 0$, both $f$ and $g$ are cubic polynomial equation (at most 10 coefficients for each). Is there any fixed method to solve this degenerate equation system? thanks.
0
votes
1answer
133 views

expanding the sqare of sum

If there any way to expand the following? $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$ and more generally, a way to expand $$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$ where $gcd(p,q) = 1$ ...
0
votes
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<...
0
votes
0answers
130 views

Random variables related through nonlinear system of equations

I asked this question on http://math.stackexchange.com/questions/377140/random-variables-related-through-nonlinear-system-of-equations, however I received no answer for a while so I'm posting it here: ...
0
votes
0answers
240 views

solving in x involving both exponential and logarithmic function

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 $b>0$?
-1
votes
1answer
505 views

equation for bowling ball on a trampoline

i´m searching for the displacement of the surface of a elastic rectangle for a given x and y and a force at a position. like a bowling ball on a trampoline the equation should include a var for the ...
-1
votes
1answer
385 views

how to solve system of congruence with multivariables [closed]

There n variables x1,x2,...,xn represented as X, n equations whose coefficient matrix (n*n) is represented as A, and this system ...