Questions tagged [multivariate-polynomial]
The multivariate-polynomial tag has no usage guidance.
6
votes
2answers
176 views
Multivariate polynomial functional equation
I’m having some difficulties solving the following functional equation:
Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which:
$P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \...
2
votes
2answers
39 views
Multivariate Quadratic Regression
I would like to make a polynomial regression, but for multivariate input data. In the univariate case, one can write polynomial regression as a multivariate linear regression problem and can come up ...
2
votes
1answer
32 views
Find random non-almost-degenerated multivariate polynomials.
If I randomly draw parameters for a polynomial of degree $n$, say $P_n$, there seems to be big chances that this polynomial can be closely approximated by a polynomial of smaller degree $P_{n-k}, k\in\...
2
votes
0answers
32 views
The Jacobian and Particular Solutions to an Underdetermined Equation
I was wondering if the factor $\sqrt{(x')^2 + (y')^2}$ in the line integral formula $$\int_a^b\ f(x,\ y)\ \sqrt{(x')^2 + (y')^2}\ dt$$ can also be thought of as a Jacobian determinant, due to the ...
1
vote
0answers
49 views
Does every nonzero polynomial take a nonzero value at one of its multi-indices?
A polynomial $p$ can be specified by its coefficient function, a finitely supported function $c:\mathbb N^d_0\to\mathbb R.$ Here $\mathbb N_0=\{0,1,2,\dots\}$ and $d\in\mathbb N_0.$ The value of $p$ ...
0
votes
2answers
80 views
Is there an efficient algorithm to find all zeros of systems of multivariate polynomial equations over a finite field?
I want my computer to solve large systems of multivariate polynomial equations over a finite field. The field is $\mathbb F_p$, where $p$ is a prime number. I heard that there is an algorithm using ...
0
votes
2answers
29 views
Representing a function in vector form.
As far as I understand, according to linear algebra, linear functions, both single and multivariable, can be represented in vector form.
For instance,
$$z = aw + bx + cy + d$$
can be rewritten as
...
1
vote
1answer
59 views
Prove that the polynomial is $g(x,y)(x^2 + y^2 -1)^2 + c$
This is from a Brazilian math contest for college students (OBMU):
Let $f(x,y)$ be a polynomial in two real variables such that the polynomials
$$\frac{\partial f}{\partial x}(x,y)$$
$$\frac{\...
2
votes
2answers
145 views
Integer solutions of a variable coefficient polynomial
I have many equations to solve similar to this one:
$$2 a b^3 - a b^2 + a b - 2 a - b^4 + b^3 - 2 b^2 + 2 b = 0$$
Here, b is a base and a is a non-zero digit in a b-adic number, so $1 \leq a \leq b-...
0
votes
0answers
23 views
Mutual Co-primeness of two multivariate polynomials
We know that in case of integers for two integers $m$ and $n$ are co-prime means we have an identity $pm+qn$=1,for some appropriate integers $p$ and $q$.I think that this happens for any Euclidean ...
0
votes
0answers
34 views
A problem regarding the defining polynomial of a projective hypersurface
I was actually trying to find out how the polynomial defining a degree $5$ projective hypersurface looks like or atleast what can be said about the polynomial provided the hypersurface satisfies some ...
1
vote
0answers
15 views
Create a multivariate function from a set of points
I am looking to construct a graph for illustrative purposes. I have a few restrictions that come from visualizing how the graph should look, in the form of $f(x,y) = z$.
$f(\infty,\infty) = \infty$
$...
0
votes
0answers
36 views
The meaning of the dimension of a Newton polytope
I am not very familar with algebraic geometry. Thus, maybe my question is a really basic one, but I have not find an answer in literature.
I consider Newton polytopes of multivariate polynomials $f(x)...
1
vote
0answers
56 views
Confusion between Wendland RBF functions - missing Wendland functions
I need to compute Wendland functions for a project, and got confused between the formula to construct Wendland functions $\phi(d,k)$ where d is the dimension
So in the original paper introducing ...
1
vote
0answers
19 views
Proving the existence of real solutions to a system of multivariate polynomials for a range of parameters
I need to prove the existence of real solution(s) to a system of 7 polynomial equalities in 14 variables up to degree 6 for a range of values of 5 real parameters $t_1,t_2,t_3,x_0,x_3$, with $t_1>0,...
0
votes
1answer
15 views
Finding Cartesian/symmetric form using $x,y$ and $z$
I am having trouble figuring out how to write a Cartesian equation for the following:
\begin{align}
x & = t\\
y & = 2t\\
z & = \cos t
\end{align}
with $0\le t \le 4\pi$. I would know how ...
0
votes
1answer
46 views
Whence Catalan's identity?
An identity attributed to Catalan is:
$A^3 = B^2 + C^2 + D^2$
where $ A = x^2 + y^2 + z^2$
$B = x(3z^2 - x^2 - y^2)$
$C = y(3z^2 - x^2 - y^2)$
$D = z(z^2 - 3x^2 - 3y^2) $ .
This can be used ...
-1
votes
3answers
76 views
Can we factor multivariate polynomial $x+xy+y$?
It seems to me we can only get $x(1+y)+y$ or $x+(x+1)y$, but maybe something better is possible with complex numbers?
It has zeros at $y=-\frac{x}{x+1}$, but how to use that fact?
4
votes
2answers
104 views
Is the ideal $J:=\langle -y^2+xz,x^5-z^3 \rangle$ prime in $\Bbb{C}[x,y,z]?$
We want to prove that the ideal $J:=\langle -y^2+xz,x^5-z^3 \rangle$
is prime in $\Bbb{C}[x,y,z].$
My attempt. An informal thought is to take the equations $y^2-xz=0,\ x^5-z^3=0 \iff y^2=xz,\ x^5=z^...
4
votes
1answer
32 views
$0 \neq \frac{f}{g}\in K(X_1,\,\dots,\,X_n)$ written as $\prod_{i=1}^n X_i^{m_i} (\frac{h}{k}),\,m_i \geq 0,\,k(0,\,\dots,\,0),\,h(0,\dots,0)\neq 0$
Let $K$ be a field. According to Grillet's Abstract Algebra (Second Edition) on page 252,
Every nonzero $\frac{f}{g} \in K(X_1,\,\dots,\,X_n)$ can be written uniquely in the form $\frac{f}{g} = X_1^...
2
votes
1answer
54 views
Proof of valuation property (function on $K(X_1,\,\dots,\, X_n)$)
Let $K$ be a field and consider the field of rational multivariate polynomials $K(X_1,\,\dots,\, X_n)$ in $K$ for some $n \geq 1$.
Define $\mathbb{P}:= (0,\, \infty)$ as the set of positive reals and ...
0
votes
0answers
73 views
Polynomial regression for the function of two independent variables
I have an old datagram with the values of some coefficients which have been retrieved experimental way.
I want to transform this pictorial representation to the form: $$z=f(x,y)$$
I believe it can ...
0
votes
1answer
31 views
Where to start studying Multivariate Polynomials
I'm a current undergrad looking to go into Theoretical Computer Science research (yes, I know about TCS Stack Exchange, but I thought the question would better fit here). In a few months, a professor ...
