Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.
6
votes
1answer
71 views
Bounding the roots of the sum of two monic polynomials with real coefficients.
Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
4
votes
1answer
155 views
Roots of bivariate polynomials
A bivariate polynomial of degree $m+n$ is,
$ p(x,y) = \sum_{k=1}^n\sum_{j=1}^m a_{jk}x^ky^j$
where $a_{mn}\neq0$ and $a_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$.
Univariate ...
1
vote
1answer
33 views
Conditions that Roots of a Polynomial be Less than Unity
Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
1answer
44 views
Skecth the root locus with respect to K for the characteristic equation.
$$s^4+12s^3+22s^2+(20+K)s+2K=0$$
I don't understand, how can I sketch the root locus? Can anybody help me to understand?
1
vote
1answer
69 views
Real Positive Zeros of Equation
During my research on physical problem, I faced the following simple equation:
$r^{2k+1}+ab\,r-a=0$
With:
$-1\leq k\leq1\:,\:0<r\:,\: a,b\in\mathbb{R}$
I need to put bounders on $a,b,k$ such ...
1
vote
1answer
38 views
Formula to scale a series that is being bent with a root / power.
I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
0
votes
1answer
310 views
Solving Polynomials in Computer Algebra Systems
Apart from low degree polynomials (2, 3, and 4) and factoring to lowest degrees, what are the method(s) used to find all the roots of a high-degree polynomial equations having only complex roots, and ...
-1
votes
1answer
84 views
Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?
When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me.
Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$.
...
-1
votes
1answer
117 views
Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots
Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors.
I'm really struggling in finding such polynomial, so ...
4
votes
0answers
51 views
How to extract roots in a complete local ring using binomial series
Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is ...
4
votes
0answers
244 views
Find roots of sum of sinusoids
Given this function and an initial point, find the next root:
$$
\begin{align}
f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
3
votes
0answers
63 views
Zeros of $ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$
What would be the zeros of the following function?
$$ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$$
This function is real and I believe it is equal ...
3
votes
0answers
53 views
Existence of roots of a polynomial equation when coefficients have varying weights
I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that:
$f_{1}(p_{1}) = f_{2}(p_{1})$.
Let ...
3
votes
0answers
26 views
Study a particular polynomial sequence
Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$
Conjecture: The sequence ...
3
votes
0answers
43 views
Existence of a Root of Elementary Monomials
Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$.
For example:
$$
m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3
$$
...
3
votes
0answers
401 views
Existence of n distinct (real) roots of an orthogonal polynomial
I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof ...
2
votes
0answers
101 views
Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?
I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and
$$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$
Let
...
2
votes
0answers
89 views
Using the duplex method to calculate square roots
I have been assigned to find out how a calculator figures out square roots, so far the shortest thing I can see is "the duplex method". But the thing is that the explanation on Wikipedia makes no ...
2
votes
0answers
76 views
How to find the root of a polynomial
I don't know how to solve the following equation: $x^5-h_1x^4-h_2x^3-h_5=0$, where $h_1=\beta_1+\beta_2$, $ h_2=\beta_1+\beta_2-\beta_1\beta_2-\frac{\beta_1\beta_2}{\beta_1+\beta_2}$, ...
2
votes
0answers
67 views
References for “closed form” numeric solutions of $\tan x=-a x$
I am looking for references that discuss solutions of the equation
$\tan x=-a x$
(for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, ...
2
votes
0answers
79 views
When does $f(x)/f'(x)$ have a first-order root?
Actually let $g(x)=0$ when $f(x)=0$ otherwise $g(x)=f(x)/f'(x)$.
Seems clear to me that if $x_0$ is an $n$-order root of $f(x)$ where $n$ is a positive integer, and $f(x)$ can be expressed as a ...
1
vote
0answers
43 views
A question about cubic equation.
I'd like to share my doubt on cubic equation.
Step 1: $ax^3+bx^2+cx+d=0$,
Step 2: We can substitute $x=y-\frac b {3a}$ to get $y^3+py+q=0$ where $p,q$ are something.
Step 3: By Vieta's ...
1
vote
0answers
29 views
Fixed Point Iteration Scheme
I have been asked to "Find a fixed point iteration scheme for minimising $f(x) = e^{cos (x)}$".
Does anybody know what a fixed point iteration scheme actually is? I know it's not Fixed Point ...
1
vote
0answers
31 views
How to solve an equation in three variables fixing two of the variables?
Also, I have the following equation, I want to solve it for $b$ keeping $a$ and $c$ fixed.
$5b^5+(60-5a)b^4+(125+50c-80a)b^3+(594c-445a-775)b^2+(2324c-1005a-3270)b+3000c-750a-3000=0.$
Also how to ...
1
vote
0answers
21 views
root of binary matrix
There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication.
Is there any way to find ...
1
vote
0answers
93 views
Given relations of coefficients and $m$ zeros of a complex polynomial, find the polynomial of degree $2n$ and $m \geq n$.
Given relations of coefficients and $m$ zeros of a complex polynomial (coefficients are complex), find the polynomial of degree $2n$ and $m \geq n$. For examples, we are finding $P(x)=C_{2n}x^{2n} + ...
1
vote
0answers
157 views
Argument Principle!!
Show that as the positive integer $N$ tends to $\infty $ , the change in argument of $e^z − z$ is bounded on 3 sides of the square with corners $ \pm 2\pi N$ $\pm 2\pi iN$ but is unbounded on the ...
1
vote
0answers
41 views
Lower-bounding the distance between zeros of a continuous function
Consider a continuous function of the form:
$L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$
where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
1
vote
0answers
117 views
Transforming root-equations into polynomials
Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples:
$\sqrt{X^4+1}$,
$\sqrt[3]{X}+\sqrt{X+1}$,
$\sqrt{X+\sqrt{X+1}}$
How can I transform a ...
1
vote
0answers
153 views
Solution to polynomial equations with non-radicals
For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial ...
1
vote
0answers
253 views
Find the first odd multiplicity root of a function
I'm trying to find the "first" (greater than some initial $t_0$) odd root (that is, a root after which the sign of the function changes) of a function $f(t)$, if there is one, that is also less than ...
0
votes
0answers
28 views
How to effciently solve a radical equation of the form $0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$?
Given a radical equation of the form
$$0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$$
where $b_j>0$ and the sign of $a_j\in\mathbb R$ matches that of $b_j-x$, is there any more efficient (analytical?) solution ...
0
votes
0answers
35 views
Unicity of solutions in several dimensions
Let $h:ℝ^{r+1}→ℝ^{r+1}$ be a real function. If $r=0$ and $h$ is bijective then we know that the equation $h(x)=y$ has a unique solution. My question is: How about the case where $r>0$? I know that ...
0
votes
0answers
149 views
Computing square roots modulo prime powers
I am trying to implement an algorithm that can compute the square root of a quadratic residue mod a prime power. For integers $a$ such that
$p\not\mid a$
$p\neq 2$
it's relatively straightforward ...
0
votes
0answers
40 views
Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.
We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$.
Let us write $L(v) \equiv L(v,p)$ ...
0
votes
0answers
46 views
Upperbound on the number of Isolated zeros of a bivariate polynomial
Let $F(x,y)$ be a bivariate polynomial, of degree n. Hence:
$F(x,y) = \underset{i+j \leq n}{\sum_{i=0}\sum_{j=0}}a_{ij}x^{i}y^{j}$
Can there exist an upperbound for the number of isolated zeros for ...
0
votes
0answers
106 views
prove cubic equation has no positive integer root
prove $q_1t^3+(k_2-1)t^2-k_2((q_1^2-1)k_1+1)^2=0$ has no Positive integer root, t is variable , $q_1$ is constant and $k_1,k_2$ are parameter
$q_1>0, k_1>0, k_2>0$, and all characters ...
0
votes
0answers
36 views
Upperbound on the number of zeros of simple continuous non-polynomials
Let $[vf(p)-g(p)]$ be a degree $n$ polynomial, one of whose roots are $p = F(v)$.
Note: We said $F(v)$ is one of the $n$ roots. Hence, it is a single valued and continuous function of $v$.
Let us ...
0
votes
0answers
37 views
Bounding number of zeros of a continuous function
Let $v$ and $p$ be related by an equation:
$vf(p) - g(p) = 0$
where $f(p)$ and $g(p)$ are polynomials of degree $n$.
Suppose we now have another polynomial in $v$ and $p$: $L(v,p)$, such that ...
0
votes
0answers
168 views
Integral of absolute value of polynomial?
Let $a(x)$ and $b(x)$ be integer irreducible polynomials where $b$ is U-shaped in the interval mentioned below and has 2 distinct real zeros. The zeros of $b$ cannot be expressed by radicals.
Also $b$ ...
0
votes
0answers
141 views
Expanding an expression using Taylor's series
We've been attempting to expand an expression with Taylor's Theorem but can't quite make the math work out.
$$
\frac{f\left(x_n\right)}{f'\left(x_n\right)}= \frac{1}{m}\frac{f^{(m)}\left(\xi ...
