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Questions tagged [multivariate-polynomial]

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6
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...