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Results tagged with polynomials
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user 9025
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
2
votes
Quick tests to differentiate eigenvalues
When I had to do this with a couple of billion matrices, I computed the traces of some powers before going for the full test. A good method is to compute $\mathrm{tr} \,((A+xI)^{2^i})$ for $i=1,2,\ldo …
- 37.7k
4
votes
The coefficient of a specific monomial of the following polynomial
This answer is really just a convoluted edition of Fedor's answer. By Cauchy's theorem,
$$f_{abc} = \frac{1}{(2\pi)^3}\int_{-\pi}^\pi \int_{-\pi}^\pi \int_{-\pi}^\pi
\frac{(e^{i\theta_1}-e^{i\thet …
- 37.7k
5
votes
Accepted
Error in Polynomial Root Finding Algorithm with Synthetic Division
It is a terrible idea to divide out roots as they are found. There will be examples where the later roots are lost almost completely. See this wikipedia article for a famous and remarkably simple exa …
- 37.7k
2
votes
Particular complex fractions
This is well known, but I'll give a short proof using 3 dimensions as an example.
Every power $x^n$ can be written as the integer linear combination of binomial coefficients $\binom xj$ for $0\le j\le …
- 37.7k
14
votes
Symmetry Group of a Polynomial
If the input has fully-expanded polynomials, then this is equivalent to graph isomorphism. … Let $P_1,P_2$ be two polynomials and $G(P_1),G(P_2)$ their corresponding graphs. …
- 37.7k
1
vote
Accepted
Finding maximum of a function with unfixed number of variables
Allow the variables to be zero, and take $k=n$.
If $a_i,a_j$ are changed to $a_i-1,a_j+1$, the function changes by $6(a_j-a_i+1)(n-2a_i-2a_j-1)$. From this, everything follows.
If there are four non …
- 37.7k
0
votes
An $n$ eigenvalue multiplicity
Let each $A_i$ be a matrix with all entries 0 except for the $(i,i+1)$ entry which is 1, where $i+1=1$ if $i=n$.
The characteristic polynomial of $\sum_j a_j A_j$ is
$x^n - \prod_j a_j$. I believe tha …
- 37.7k
6
votes
A question on the real root of a polynomial
(This is a comment, not an answer.)
If $f_n(x)$ is your polynomial, starting with $f_0(x)=1$, then
$$ \sum_{n=0}^\infty f_n(x) y^n =
\frac{1-xy+x^2y^2+x^2y^3}{(1+xy^2)(1-xy-xy^2)}
= 1 + \frac{x …
- 37.7k
25
votes
3
answers
1k
views
Changing the signs of the coefficients of a polynomial to make all the roots real
This question might just possibly (on a good day) be of relevance to a problem on graph polynomials. …
- 37.7k
2
votes
Accepted
Closed-Form solution for system of simple nonlinear equations
It isn't clear to me that the solution is unique, even with the positivity restriction.
But, anyway, one practical method that sometimes works is to find an iteration that converges. My first attempt …
- 37.7k
1
vote
$\prod_k(x\pm k)$ in binomial basis?
Let $F(x)$ be the right side. You need that $F(x)=0$ for [corrected] $x\in\{-n,-n+1,\ldots,n-1,n\}$. Assume we have one of those $x$s. The summation can stop at $m=n-x$ as later terms are zero. For …
- 37.7k
1
vote
Polynomials all of whose roots are rational
Robert Israel's comment about using Sturm sequences got me thinking about how it could be done using only the original polynomial $f(x)$ (assumed to have distinct rational zeros). If the degree is odd …
- 37.7k
9
votes
On the polynomial $\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$
The divisibility is easy to prove and a more general phenomenon. Let $Y=1-X$, then
$$F_n(X) = \sum_{k=0}^n (-1)^k \binom nk (1-Y)^{k(n-k)}=\sum_{r\ge 0} (-1)^r a_{r,n}Y^r$$ where $$a_{r,n} = \sum_{k= …
- 37.7k
3
votes
How different can the constituents of an Ehrhart quasi-polynomial be?
However the difference between the polynomials for even and odd dilations has degree only 5. I don't know (but would like to know) what happens for larger matrices. …
- 37.7k
4
votes
Degree necessary of a polynomial?
The solutions look very much like scaled-and-shifted Chebyshev polynomials inside the box and surely that is the place to look for an analytic upper bound on the degree.
Here is my method. …
- 37.7k