Questions tagged [algebraic-equations]
The algebraic-equations tag has no usage guidance.
41
questions
12
votes
3
answers
2k
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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 ...
11
votes
3
answers
651
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
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \...
10
votes
2
answers
625
views
When is the degree of this number 3?
I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that ...
9
votes
1
answer
913
views
Impact of Ramanujan's Note on a set of simultaneous equations
I had been pointed to Ramanujan's 1912 article Note on a set of simultaneous equations in this answer to my former question about the Solvability of a system of polynomial equations.
While the ...
7
votes
1
answer
1k
views
Can roots of any polynomial be expressed using Eulerian function?
I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld:
$$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$
It is interesting because it is claimed that roots of any ...
7
votes
2
answers
486
views
Bivariate polynomials with special properties
I recently came across some polynomials with some remarkable properties.
A polynomial $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if
the set of solutions to the system $P(u,v)=P(v,u)=0$...
6
votes
1
answer
2k
views
Is there an analytical method of solving general square root equations?
Equations such as $\sqrt{x+1}+\sqrt{x+2}=x+3$ are easily solvable by squaring both sides. But if we increase an extra square root, like if trying to solve $\sqrt{x+1}+\sqrt{x+2}+\sqrt{x+3}=x+4$ we ...
5
votes
5
answers
826
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For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
5
votes
1
answer
312
views
Is it possible to solve sextic equations using the Fox H function?
Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago.
In contrast, we know more about the Fox H function, and we ...
5
votes
1
answer
436
views
Abel-Ruffini theorem for systems of polynomial equations
I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...
5
votes
0
answers
129
views
Selberg zeta function analytic expressions
Consider the following algebraic equation,
$$
y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}
$$
which is a Riemann surface of genus $n-1$ (after compactifying). The classical retrosection theorem due to ...
4
votes
2
answers
411
views
Roots of polynomials of particular type
How to find the solutions $x $ of the following equation: $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.
For the case $k=2$, I get ...
4
votes
1
answer
969
views
Equations of the Hirzebruch surface embedded in a large space.
Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$ and let $D$ be the very ample divisor $3C_0+5f$ on $\mathbb{F}_1$ (notation as in [Hartshorne, Algebraic ...
4
votes
1
answer
304
views
Does this system have a closed-form solution? $x_j^2 = \sum_{i=1}^n B_{ij} x_i$
I am interested in solving the following system of $n$ equations:
$$x_j^2 = \sum_{i=1}^n B_{ij} x_i $$
for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer and all the $0\leq B_{ij}\leq 1$ ...
4
votes
2
answers
503
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Closed-Form solution for system of simple nonlinear equations
I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...
4
votes
0
answers
2k
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how to solve this multivariate quadratic equation?
This has been posted on math.stackexchange but got just one partial(insightful though) comment. I'm posting it here in a hope of getting further ideas and comments:
The problem was:
Any hope to ...
3
votes
1
answer
223
views
Existence of solution for this set of polynomial equations
We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where
$$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$
I'm interested in proving that a solution for ...
3
votes
1
answer
288
views
Solvability of a system of polynomial equations
What can be said about the solutions of, resp. solving, the following system polynomial equations, in which the $x_i$ and $y_j$ are the variables and $c_{i,j},\,d_i\in\mathbb{R}$ are constants:
$$\...
3
votes
1
answer
211
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 ...
3
votes
0
answers
227
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
2
answers
524
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 ...
2
votes
3
answers
721
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
$$0 = 1 + \sum_j \left(t_j^{2l+1} + {t_j^*}^{2l+1}\...
2
votes
2
answers
178
views
Roots of modified polynomials
Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...
2
votes
0
answers
613
views
System of two variables quadratic equations
Let $\mathcal{P}_{2,Z}$ be the set of all 2 variables quadratic equations $P(x,y)$ with integral coefficients: $$P(x,y)=a_1x^2+a_2y^2+a_3xy+a_4x+a_5y+a_6\ \ \ \ \ \ (a_i\in \mathbb{Z})$$
Consider a ...
1
vote
1
answer
98
views
Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints
Let $\mathbb{D}$ be the open disc. It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a ...
1
vote
1
answer
161
views
Algebraic solution for a system of algebraic equations?
How would one solve algebraically the following system of algebraic equations?
$$f(a,b):=a(1-b)+ab\frac a{a+b}.$$
$$u = f(a,b),\quad v = f(b,a).$$
Solve algebraically $(a,b)$ in terms of $(u,v)$
...
1
vote
2
answers
4k
views
Real root of a cubic equation [closed]
I have a function f(x,n) can be expressed as a cubic function of x with coefficients that are functions of n. For example x^3 + (n-2)x^2 + (3n-6)x + n.
I want to prove that for every positive value ...
1
vote
0
answers
39
views
Existence of real solutions to nonlinear algebraic equation: conditions on coefficients
Good day. I am dealing with the following system of nonlinear algebraic equations:
$$
x_j = \prod_{k=1}^N (1 + x_k)^{A_{j,k}}\,,\quad j=1,\ldots,N\,,
$$
where $A_{j,k}\in\mathbb{Z}$.
I would like to ...
1
vote
0
answers
102
views
Roots of the system of quadratic equations
For a set of $m$ positive semi-definite $d\times d$ matrices $Q_j$ I have the following system of equations over column-vector $\vec{x}$:
$$
q_j = \vec{x}^T Q_j \vec{x}, \quad j=1,\dots,m
$$
with ...
1
vote
0
answers
113
views
Algebraic relation amongst an elliptic function and its convolution
NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett.
I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
1
vote
0
answers
43
views
Solving exponential system [closed]
How can i solve the following system?
$
\begin{cases}
x^y = 16 \\
\frac{x}{y}=2\\
\end{cases}$
I tried everything, nothing works.
1
vote
0
answers
74
views
About the rank of a Pell equation-related matrix
I have a question about the solution of Pell-equation over a prime field.
I want prove that the matrix $M$ is of rank $\frac{p-1}{2}$, with $M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} \right)^{(...
1
vote
0
answers
148
views
How to solve a system of quadratic equations with multiple unknowns? [closed]
While solving a problem i have ended up with these 4 equations:
a^2+b^2= p;
(c-q)^2+d^2=r;
(d-b)/(c-a)=s;
(c-a)^2-(d-b)^2=t
Here a,b,c,d are unknowns . The rest (p,q,r,s,t) are known values. I am ...
1
vote
0
answers
170
views
satisfiable polynomial equations for given free coefficients
Let $F$ be a finite field, $n, k, m$ be natural numbers.
I give you $m$ vectors $c^{(1)},\ldots,c^{(m)}\in F^n$.
I ask for polynomials $p_1,\ldots,p_n$ on $k$ variables over $F$ such that the system ...
0
votes
1
answer
480
views
Pair of two-variable cubic polynomial equations
Let us consider the following system of two polynomial equations
of third order for two real numbers $x_1,x_2$:
$$x_i (x_i + 2) (x_i + 4) - 2 a_i (x_1 + x_2 + 4) = 0,$$ $i =1,2$.
Here $a_1 >0$ ...
0
votes
3
answers
2k
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Is there any method to solve a Bivariate Cubic Equation System? [closed]
$f(x, y) = 0$ and $g(x, y) = 0$,
both $f$ and $g$ are cubic polynomial equation (at most 10 coefficients for each).
Is there any fixed method to solve this degenerate equation system?
thanks.
0
votes
2
answers
76
views
Non-negative integer solutions of a system of equations $\sum_{i=1}^{n} x_i^2 = 4k-6, \sum_{i=1}^{n} x_i = 2k$
Fix $k \ge 3$, $n \ge 2k$. Consider the following system of equations:
\begin{align}
\sum_{i=1}^{n} x_i^2 = 4k-6, \\
\sum_{i=1}^{n} x_i = 2k.
\end{align}
It seems that the only non-negative integer ...
0
votes
1
answer
344
views
Solving an equation involving $x$ both squared and inside a logarithm [closed]
Is it possible to solve a function with both exponential and logarithm such as
$$
a x^2 - b\cdot\log(x) = c
$$
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
0
votes
0
answers
255
views
Random variables related through nonlinear system of equations
I asked this question on https://math.stackexchange.com/questions/377140/random-variables-related-through-nonlinear-system-of-equations, however I received no answer for a while so I'm posting it here:...
-1
votes
1
answer
313
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Expanding the square of sum [closed]
If there any way to expand the following?
$$\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 $\gcd(p,q) = 1$
...
-1
votes
1
answer
792
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 ...