# Questions tagged [algebraic-equations]

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### Solving a system of quadratic equations in nine variables [migrated]

I'm trying to carry out the forward displacement analysis of a parallel mechanism. This question is simplified as follows. Let $Q_1$, $Q_2$, $Q_3$, $A_1$, $A_2$, $A_3$ be six unit vectors that pass ...
1 vote
54 views

### Combine ODE with constraints using a Lagrange multiplier [migrated]

Consider a constrained ODE system: \begin{align} \dot{\bf x} &= \bf f(t,\bf x), \\ st. 0 &= \bf g(t,\bf x). \end{align} I wish to combine these into a single equation using a Lagrange ...
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### Is it possible to solve sextic equations using the Fox H function?

Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago. In contrast, we know more about the Fox H function, and we ...
• 171
830 views

### Impact of Ramanujan's Note on a set of simultaneous equations

I had been pointed to Ramanujan's 1912 article Note on a set of simultaneous equations in this answer to my former question about the Solvability of a system of polynomial equations. While the ...
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### Selberg zeta function analytic expressions

Consider the following algebraic equation, $$y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}$$ which is a Riemann surface of genus $n-1$ (after compactifying). The classical retrosection theorem due to ...
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1 vote
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### Roots of the system of quadratic equations

For a set of $m$ positive semi-definite $d\times d$ matrices $Q_j$ I have the following system of equations over column-vector $\vec{x}$: $$q_j = \vec{x}^T Q_j \vec{x}, \quad j=1,\dots,m$$ with ...
• 171
1 vote
101 views

### Algebraic relation amongst an elliptic function and its convolution

NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett. I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
• 95
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### Roots of polynomials of particular type

How to find the solutions $x$ of the following equation: $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers. For the case $k=2$, I get ...
• 1,129
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### Non-negative integer solutions of a system of equations $\sum_{i=1}^{n} x_i^2 = 4k-6, \sum_{i=1}^{n} x_i = 2k$

Fix $k \ge 3$, $n \ge 2k$. Consider the following system of equations: \begin{align} \sum_{i=1}^{n} x_i^2 = 4k-6, \\ \sum_{i=1}^{n} x_i = 2k. \end{align} It seems that the only non-negative integer ...
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### 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 $g$, ...
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1 vote
140 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 ...
204 views

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 ...
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### 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.
716 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 ...
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 ...
• 123
1 vote
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 ...
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### 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:...
• 267
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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 ...
688 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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437 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 $P(u,v)=P(v,u)=0$...
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