Questions tagged [algebraic-equations]

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12 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 ...
user8308's user avatar
  • 121
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} \...
Andrej Bauer's user avatar
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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 ...
Ricky's user avatar
  • 3,674
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 ...
Manfred Weis's user avatar
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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 ...
Anixx's user avatar
  • 9,306
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 $P(u,v)=P(v,u)=0$...
Per Alexandersson's user avatar
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 ...
baronbrixius's user avatar
5 votes
5 answers
826 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 ...
Taicho's user avatar
  • 225
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 ...
Ember Edison's user avatar
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 ...
curiousStudent's user avatar
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 ...
Sounak Sinha's user avatar
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 ...
GA316's user avatar
  • 1,219
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 ...
gio's user avatar
  • 1,149
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 $0\leq B_{ij}\leq 1$ ...
user_lambda's user avatar
4 votes
2 answers
503 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 ...
Mario Krenn's user avatar
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 ...
yuanz07's user avatar
  • 123
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 ...
R B's user avatar
  • 608
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: $$\...
Manfred Weis's user avatar
  • 12.6k
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 ...
loup blanc's user avatar
  • 3,574
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-...
მამუკა ჯიბლაძე's user avatar
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 ...
user64494's user avatar
  • 3,309
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}\...
nik's user avatar
  • 23
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 $g$, ...
Felix Goldberg's user avatar
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 ...
Kim Jun-Lee's user avatar
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 ...
GBA's user avatar
  • 125
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 $(u,v)$ ...
Hans's user avatar
  • 2,169
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 ...
Austen's user avatar
  • 13
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 ...
Stefano's user avatar
  • 105
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 ...
Krivoi's user avatar
  • 171
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 ...
Stefano's user avatar
  • 105
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.
Canteri Marco's user avatar
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)^{(...
Diane's user avatar
  • 71
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 ...
Mehrab Shahriar's user avatar
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 ...
user17119's user avatar
  • 179
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 $a_1 >0$ ...
Vladimir's user avatar
  • 359
0 votes
3 answers
2k 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.
user37959's user avatar
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 ...
Jianrong Li's user avatar
  • 6,101
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 $b>0$?
pratikag's user avatar
  • 109
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:...
Tomas's user avatar
  • 267
-1 votes
1 answer
313 views

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 $\gcd(p,q) = 1$ ...
xwangae's user avatar
  • 191
-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 ...
Ecolss's user avatar
  • 1