# Questions tagged [algebraic-equations]

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33
questions

**1**

vote

**1**answer

126 views

### Algebraic solution for a system of algebraic equations?

How would one solve algebraically the following system of algebraic equations?
$$f(a,b):=a(1-b)+ab\frac a{a+b}.$$
$$u = f(a,b),\quad v = f(b,a).$$
Solve algebraically $(a,b)$ in terms of $(u,v)$
...

**5**

votes

**1**answer

223 views

### Abel-Ruffini theorem for systems of polynomial equations

I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...

**0**

votes

**1**answer

208 views

### Pair of two-variable cubic polynomial equations

Let us consider the following system of two polynomial equations
of third order for two real numbers $x_1,x_2$:
$$x_i (x_i + 2) (x_i + 4) - 2 a_i (x_1 + x_2 + 4) = 0,$$ $i =1,2$.
Here $a_1 >0$ ...

**4**

votes

**2**answers

353 views

### Closed-Form solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...

**4**

votes

**1**answer

258 views

### Does this system have a closed-form solution? $x_j^2 = \sum_{i=1}^n B_{ij} x_i$

I am interested in solving the following system of $n$ equations:
$$x_j^2 = \sum_{i=1}^n B_{ij} x_i $$
for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer and all the $0\leq B_{ij}\leq 1$ ...

**1**

vote

**0**answers

38 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.

**3**

votes

**2**answers

283 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 ...

**1**

vote

**0**answers

61 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)^{(...

**3**

votes

**0**answers

216 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-...

**3**

votes

**1**answer

183 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 ...

**12**

votes

**3**answers

568 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} \...

**2**

votes

**1**answer

121 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 $g$, ...

**1**

vote

**0**answers

135 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 ...

**3**

votes

**1**answer

191 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 ...

**0**

votes

**3**answers

1k 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.

**-1**

votes

**1**answer

649 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 ...

**4**

votes

**0**answers

2k 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 ...

**1**

vote

**2**answers

4k 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 ...

**0**

votes

**0**answers

186 views

### Random variables related through nonlinear system of equations

I asked this question on https://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:...

**2**

votes

**0**answers

598 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**

votes

**1**answer

638 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 ...

**2**

votes

**1**answer

618 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 ...

**6**

votes

**1**answer

1k 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 ...

**2**

votes

**3**answers

675 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}\...

**0**

votes

**0**answers

257 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 $b>0$?

**9**

votes

**2**answers

537 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 ...

**1**

vote

**0**answers

242 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 ...

**0**

votes

**1**answer

137 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 $gcd(p,q) = 1$
...

**7**

votes

**1**answer

896 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 ...

**5**

votes

**5**answers

785 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 ...

**1**

vote

**0**answers

164 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 ...

**7**

votes

**2**answers

392 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 $P(u,v)=P(v,u)=0$...

**11**

votes

**3**answers

2k 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 ...