All Questions
Tagged with systems-of-equations polynomials
514 questions
-2
votes
0
answers
33
views
Find equation(s) that satisfies all requirements [closed]
Which equation(s) for y(x) satisify these requirements?
y(0)=0
y(x1)=y2
y(x_max)=y_max
y'(x_max)=0
y'(x)>0 for 0<x<x_max
y''(x)<0 for 0<x<x_max
Example numbers:
(x1, y2) = (2, 7)
(...
- 1
-1
votes
1
answer
81
views
Solving polynomial system of equations [closed]
Context: I have to solve the following system of three equations that are in descending powers of $\alpha$:
$$\begin{cases}0&=&A\alpha^2+B\alpha+C\\ 0&=&\frac13 A\alpha^3+\frac12 B\...
- 1
-1
votes
1
answer
46
views
Finding a Polynomial Equation in $x$ and $y$. Given $x=a+b$ and $y=ab$.
I am working with the relationships $x=a+b$ and $y=ab$, where $a,b,x$ and $y$ are real variables. I would like to know if it is possible to derive any polynomial equation $p(x,y)=0$ that relates $x$ ...
- 71
4
votes
1
answer
83
views
About a functional equation
Problem: Let $(f, g)$ be a pair of function that satisfies the following:
$f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$
$\forall x, y \in \mathbb{R}^+, f(xy + f(x)) = xf(y) + g(x)$
Find all possible ...
- 141
3
votes
1
answer
467
views
Need help with a system of 2 very verbose equations
I need to find a $\gamma$ in terms of $a$ and $r$ which solves
$$(a^2+ r^2-2 ar\cos\beta)+(a^2+\gamma^2 r^2-2a\gamma r\cos\beta) - 2\sqrt{(a^2+\gamma^2 r^2-2\gamma ar\cos\beta)(a^2+r^2-2ar\cos\beta)} =...
- 763
2
votes
1
answer
64
views
Is every finite collection of points in $\mathbb{C}^n$ the solution to a compatible algebraic system?
Preliminaries:
Bezout's theorem tells us that if we have a system of $n$ multivariate polynomials over $\mathbb{C}$
$$P_1(x_1 .. x_n) = 0 \\ \vdots \\ P_n(x_1 ... x_n) =0 $$
Of degrees $d_1 ... d_n$ ...
- 17.9k
0
votes
0
answers
29
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Finding a relation between l, m, n given $a_1l+b_1m+c_1n=d_1$ and $a_2l^2+b_2m^2+c_2n^2=d_2$
In quite a few analytic geometry questions, we need to find the relation between l,m and n given the set of equations :
$$ a_1l+b_1m+c_1n=d_1 $$
$$a_2l^2+b_2m^2+c_2n^2=d_2$$
Is there a general ...
- 137
0
votes
0
answers
46
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Lower bound for distance between roots of polynomial equations
Consider a set of multivariate polynomial equations, i.e. $f_i(x_1, \dots, x_n)=0$ for $i=1,\dots,m$ with $m,n\in\mathbb{N}$ and $f_i$ a polynomial. Assume the set of solutions of these equations is ...
- 1
1
vote
0
answers
24
views
Given a system of polynomials over $F_2$, find the element which has the maximum number of solutions.
Given a system of polynomials over a binary polynomial ring $F_2[x_1,...,x_n]$ $$S = [f_1(x_1,...,x_n),f_2(x_1,...,x_n),...,f_n(x_1,...,x_n)] $$.
I want to find the element $y \in F_2^n$ which has the ...
- 403
0
votes
0
answers
63
views
Constructing a System of Two Cubic Polynomial Equations with Exactly 9 Real Solutions in Maple
I am trying to construct a system of two cubic polynomial equations in two variables (x and y) with exactly 9 real solutions using Maple. However, I am having trouble finding the appropriate ...
- 23
0
votes
0
answers
25
views
Multivariate quadratic system of equations with special structure
Given $A_1,...,A_5$ all rank-1 matrices in $\mathbb{R}^{3\times3}$, consider the following system of equations:
$$ v_1^\top A_i v_1 + v_2^\top A_i v_2 = 0 \quad \forall i \in\{1, .., 5\}$$
Where $v_1$ ...
- 121
1
vote
3
answers
187
views
Solve for real $x,y,z$ : $x^2 + xy + y^2 = a$, $y^2 + yz + z^2 = b$, $z^2 + xz + x^2 = c$ .
Solve for complex $x,y,z$ : $x^2 + xy + y^2 = a$, $y^2 + yz + z^2 = b$, $z^2 + xz + x^2 = c$ where $a,b,c \in R$ such that $a,b,c \ge 0$ .
We've :
$x^2 + xy + y^2 = a \dots(1)$
$y^2 + yz + z^2 = b \...
- 1,258
3
votes
2
answers
138
views
Solving the system $x^4+y^4+z^4=a$, $xy+xz+yz=b$, $xyz=c$
I am trying to solve the following system of equations:
$$
\begin{cases} x^4+y^4+z^4=a\\[4pt] xy+xz+yz=b\\[4pt] xyz=c\end{cases}
$$
where $a$, $b$ and $c$ are constants and $x$, $y$ and $z$ are the ...
- 31
0
votes
0
answers
23
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A function of specific form passing through two given points
Let
$$s(t; a_0)=a_{0}t^{2}\left(\frac{1}{2}-\frac{t}{3T(a_0)}\right)$$
with $T(a_0)=\sqrt{\frac{6d}{a_{0}}}$ (where $d$ is some positive real constant).
Then, let
$$
s^*(t; a_0, t_w) = s\left(\frac{t-...
0
votes
1
answer
52
views
approximate solution of polynomial equation
I am trying to solve the Following equation for r,
$$2 a Q^4+5 r^4 \left(3 c (\omega +1) r^{1-3 \omega }-2 r (r-3 M)-4 Q^2\right)=0$$
Clearly this is unsolvable. But if we substitute a=0 and c=0, the ...
1
vote
1
answer
53
views
How to solve, or quantify solutions of, polynomial equations in $\mathbb{F}_2[x,y]/\langle x^\mu - 1, y^\nu - 1\rangle$? [closed]
Suppose I was given an equation in $\mathbb{F}_2[x,y]$ under the identification $x^\mu = 1$ and $y^\nu = 1$ for some integers $\mu,\nu$, with some unknowns $c[x,y]$ and $d[x,y]$. For example:
\begin{...
- 207
0
votes
2
answers
80
views
how to prove $\forall a_n \in \mathbb{R}, n\in \mathbb{N} \exists x \in \mathbb{R} : \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $?
I tried to prove that $\forall a_n \in \mathbb{R} , n\in \mathbb{N}$ then there exist a real root $x$ such that
$$ \sum_{k=1}^n a_k \left( x^k-\frac{1}{k+1} \right)=0 $$
for example if $n=2$ and $a_1=...
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-1
votes
1
answer
88
views
Find the equation of the line(s) simultaneously tangent to $y = −x^4$ and $y = (x + \frac{5}{2})^4 + 16$
This is a A-level further maths question. To find a line which is simultaneously tangent to both quartic equations
- 1
0
votes
0
answers
38
views
Method for solving polynomial system without multilinear form?
I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
-1
votes
3
answers
109
views
If $ a^2 + b^2=1,c^2+d^2=1,ac+bd=0 $ , find $ab+cd $?
I have tried everything except trigonometry.We have not still started doing trigonometry in classes.
I don't know how am I supposed to solve this,since I put 5 hours and nothing led to solution.
- 33
0
votes
0
answers
33
views
How do I get $c$ and $d$ parameters?
I am starting the Harold M Edward's book "Galois Theory". The first sections explain the second, third and fourth grade polynomic equations and I got stucked in this exercise. I don't know ...
- 21
1
vote
2
answers
98
views
Why is this particular substitution made? ($y=tx$)
A problem in Complex Numbers (Andreescu, Andrica):
Solve the equation $z^3 = 18 + 26i$, where $z = x + yi$ and $x$, $y$ are
integers.
Solution given :
We can write $(x + yi)^3 = (x + yi)^2 \times (...
- 137
3
votes
3
answers
156
views
Solve for $x$ and $y$ given $xy + x^2 + y^2 = 109$ and $x^2-y^2=24$
I was able to solve this graphically but algebraically I'm lost. The two equations are:
$xy + x^2 + y^2 = 109$
$x^2-y^2=24$
What I've tried: substitute equation (2) into (1) to get:
$$
2y^2+xy-85=0
$...
- 205
0
votes
0
answers
27
views
How to solve multivariate polynomial equation on multivariate polynomial ring
Is there a general approach to solve a $f(x) \in R[x]$ which satisfy an equation like $a_0(x)+a_1(x)f(x)+a_2(x)f^2(x)+... = 0$ where $a_i(x)\in R[x]$, $R$ is a ring.
Further, is there a general ...
- 33
1
vote
2
answers
85
views
Semi-numeric solutions to a system of polynomial equations when a Groebner basis is too complicated
I have a system of polynomial equations with rational coefficients and I would like to find real solutions, if they exist.
The system has $n\sim 10$ unknowns, $n$ equations with degree $\sim 2n$ and ...
- 4,914
0
votes
1
answer
38
views
Using a system of non-linear equations to prove an identity
Prove that if $3xy + 2yz + z + 1 = 0$ and $3zx + 2z + x + 1 = 0$, then $3xy + 2x + y + 1 = 0$.
I think the solution will involve combining these two equations in some way. I have attempted many ...
- 1,970
2
votes
1
answer
99
views
Proportionality of a system of polynomials
I am currently reading Ivanovs „Easy as Pi?“ in fact, I am trying to understand a proof in this book. Following statement is to prove:
For $n\geqslant 3$ the curve $x^n+y^n=1$ has no rational ...
5
votes
1
answer
163
views
A system of polynomials has Galois group G, a subgroup of $P_n$. Why are invariant polynomials of the roots rational, can you calculate them?
I have been studying systems of equations based upon iterating a polynomial. The complete Galois group of these systems is only a subgroup of the permutation group. There are many invariant ...
- 634
2
votes
4
answers
300
views
Roots of $ 16 x^5 - 20 x^3 + 5x + 1 = 0 $
The following is from Edexcel further mathematics Core Pure Book 2 A Level Mixed Exercise 1 Question 9 part b:
9 a Use De Moivre's Theorem to show that
$$ \cos 5\theta \equiv 16 \cos^5 \theta - 20\...
- 22.6k
1
vote
1
answer
80
views
What's the name for this method of putting the roots of a polynomial in a system of equations?
As I was (and still am lol) struggling on the exercises for symmetric polynomials in my abstract algebra book, I stumbled upon this neat relating the solutions of a polynomial in $\mathbb C$ with the (...
- 932
0
votes
0
answers
57
views
How to calculate the min and max points of an ellipse
The question is inspired by: Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers..
The first way to solve the equation is by solving the quadratic equation for $x$ - $x_{1,2}=\frac{...
- 580
2
votes
2
answers
46
views
Can the implication $(x_1 = 0) \rightarrow (p(x_1,...,x_n) = 0)$ be encoded in a system of polynomial constraints in $\mathbb{C}[x_1,...,x_n]$?
Consider a set $S$ of polynomials in $\mathbb{C}[x_1,x_2,...,x_n]$, the polynomial ring of $n$ variables over the complex numbers.
The set $S$ can then be interpreted as a system of constraints on the ...
- 902
2
votes
1
answer
168
views
Solving the system $x+y+z=1$, $x^2+y^2+z^2=2$, $xyz-xy-xz-yz=3$
$$\begin{align}
x+y+z&=1 \\
x^2+y^2+z^2&=2 \\
xyz-xy-xz-yz&=3
\end{align}$$
My Attempt:
$(x+y+z)^2=x^2+2xy+2xz+y^2+2yz+z^2=1$
$xyz-xy-xz-yz=3$
$xy+xz+yz+3=xyz$
$xyz-3=xy+xz+yz$
$1=x^2+2xy+...
- 85
1
vote
1
answer
205
views
How to determine if an equation is linear or quadratic relationship?
I have an equation of the form
$y= \frac{v}{x^2} - \frac{1}{x}$. I can rewrite it as $y= vx^{-2} - x^{-1}$. I don't know if it's quadratic. Any suggestions on what this relationship would like?
- 11
5
votes
3
answers
562
views
A tricky system of non-linear multivariate equations
Given $$\begin{align*}
x^2 + y^2 &= z^2\\
ax + by &= cz\\
a^2 + b^2 &= c^2\\
\end{align*}$$
how can I solve for $y$ and $z$ in terms of $x$?
My work so far:
Note that once we solve for $y$...
- 5,275
1
vote
1
answer
74
views
Can we prove the following equality?
Let $n$ be an odd integer and $a_1, \cdots, a_n$ distinct complex numbers. Suppose that for each $1 \leq i \leq n$, the set $$( \dfrac{a_j}{a_i}: 1 \leq j \leq n, j \neq i )$$
consists of $\dfrac{n-1}{...
- 11
3
votes
0
answers
77
views
How can I "solve" or simplify quadratics of two variables?
Given a multivariate polynomial equation like $$xy - bx - ay - ab = 0$$
how do I simplify it, make sense of it, or characterize it?
My attempts so far are below.
Attempt 1
Viewing $x,y$ as variables ...
- 5,275
4
votes
1
answer
262
views
Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?
I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
- 43
0
votes
0
answers
42
views
number of solutions for an underdetermined quadratic polynomial system
For an underdetermined quadratic polynomial system like the one given below, how does one determine the number of solutions?
$a^2+4bd=0$
$-2af+4(be+cd)>0$
$f^2+4ce=0$
where $(a,b,c,d,e,f)$ are the ...
- 39
0
votes
1
answer
46
views
Question from elementary algebra I'm having trouble with
If 1/3 be added to the numerator of a certain fraction the fraction will be increased by 1/21, and if 1/2 be taken from the denominator the fraction becomes 8/9: find it.
...
1
vote
0
answers
104
views
Finding the number of solutions of a system of multivariate polynomials without solving the system.
I have a system of multivariate polynomial equations, say for 3 variables,
$$
f_1(x,y,z)=0 \ , \quad f_2(x,y,z)=0 \ , \quad f_3(x,y,z)=0
$$
I need to find the number to solutions to this problem. I am ...
- 315
3
votes
0
answers
76
views
doubt with an equation
I was reading this exercise:
If $x=\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}$, then the value of $x$ is?
Solution:
When $a+b+c\not =0$, from the given equalities we have
$$a=(b+c)x, \,\,\, b=(a+c)x\, \...
1
vote
1
answer
92
views
How to isolate one of the terms or derive an expression to arrive at one of them from given expressions
So I have been given 4 expressions and in three variable $a,b$ & $c$. Using these three expressions I have to isolate either of $a,b$ or $c$. However, I have not been able to come up with any ...
- 177
0
votes
1
answer
67
views
Textbooks on basic algebra (with exercises)
I am looking for textbooks (ideally with a lot of exercises) on basic algebra.
In particular, I am interested in the following topics:
basic calculations with real and complex numbers
use and ...
- 123
1
vote
1
answer
100
views
How to solve this system of multivariate polynomial equations for $0<x_7<x_6<x_8 \le 1$? Groebner basis maybe?
I am reformulating my question according to the guidelines I was given.
I have the following problem: I cannot find a way to solve the system of equations further down. This is the calculations from ...
- 31
1
vote
1
answer
167
views
Why homogeneity of an equation is preserved even when we change variables?
Consider the equations,
$$x^{2}+y^{2}+z^{2}-xt-t^{2}=0 \tag{1}$$
$$x^{2}+y^{2}+z^{2}+yt-2t^{2}=0 \tag{2}$$
Clearly, both equations are homogeneous. Solve for $t$ from the above equations. You will get ...
- 545
0
votes
1
answer
561
views
How can we find the coefficients of a polynomial given the coordinates of 3 points?
I tried solving the system of $3$ equation for this, and it's actually very hard. I'm just trying to find the coefficients $A,B,C$, of the polynomial $Ax^2+Bx+C$ given the points $(a,b) (c,d) (e,f)$...
- 333
2
votes
1
answer
93
views
How to pick the "correct" solution from a system of polynomial equations?
I have a system of equations of the form
$$
\begin{aligned}
a &=z+\frac{x}{y^2} &(1)\\
b &=x-2c\frac{x}{y}+ac^2 &(2)\\
c &=y-\frac{z}{x}y^3 &(3),
\end{aligned}
$$
where $a>0$...
- 292
0
votes
5
answers
225
views
Finding redundant equations in a underdetermined system of multivariate polynomial equations over $\Bbb R$
Starting from a geometric problem, I came up with a system of multivariate (many lines and points) polynomial equations where some equations are redundant (because they correspond to redundant ...
- 145
2
votes
1
answer
153
views
If $a,b,c,d$ are the roots of the equation biquadratic equation $x^4+px^3 +qx^2+rx+s=0$ , find the value of $\Sigma a^2b^2$.
If $a,b,c,d$ are the roots of the equation biquadratic equation $x^4+px^3 +qx^2+rx+s=0$
, find the value of $\Sigma a^2b^2$.
My solution goes like this:
Since, $a,b,c,d$ are the roots of the equation ...
- 2,630