Questions tagged [multivariate-polynomial]

Let $R$ be a ring. A multivariate polynomial $p(x_1,\ldots,x_n)$ over $R$ is a finite sum of powers of the $x_i$s multiplied by coefficients in $R$.

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If the real zeroes of real polynomial p(x,y) are disjoint points and curves, is p(x,y) a positive sum of squares?

For example, $p(x,y) = x^2(x-1)^2 + y^2(y-1)^2$ has real zeroes in the set $\{(0,0), (0, 1), (1, 0), (1, 1)\}$ and admits a decomposition into a sum of squares. How can I find decompositions like ...
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Simple multivariate orthogonal polynomials over the simplex

This is a soft question; I am looking for the most straightforward constructions of an orthogonal family of polynomials satisfying the following: multivariate (say in two variables), symmetric (if $p(...
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How does one write the factorization of a general multivariable polynomial?

For a general univariate polynomial of order $n$, $P_n(x)$, I can express it succinctly as $$ P_n(x) = C\prod_{i = 1}^n (x - r_i), $$ where $C$ is a constant and $r_i$ are the roots. Is there a ...
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Is there a notion of transcendental numbers for polynomials of multiple variables?

We say $\alpha\in\mathbb{C}$ is algebraic if there is a polynomial $p(x)\in\mathbb{Z}[x]$ such that $p(\alpha)=0$, and we say $\alpha$ is transcendental if no such polynomial exists. Examples of ...
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What does it mean for two equations to be 'invariant'?

I have watched a few videos yet I am still having a bit of trouble. I have included the two equations here: It says that the two equations are invariant and must be anchored. I am totally lost on ...
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63 views

Help Solving Multivariable Quintic Function

I am developing a game, and I am working out some of the equations for one of the features of it. Essentially, I am adding a way for a character's skills to be inherited when they die, and this ...
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37 views

Irreducible subsets of the affine plane [duplicate]

Here's the problem I'm working on: Let k be an algebraically-closed field, and let $X \subseteq \mathbb{A}^2$ be a closed, irreducible set. Show that either: $X = Z(0)$, i.e. $X = \mathbb{A}^2$ $X = ...
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Multivariate Quadratic Regression For 3+ input variables

A similar question has been asked here: Multivariate Quadratic Regression, but my question is, how do you take the same matrix: $$ \pmatrix{N &\sum u_i &\sum v_i & \sum u_i^2 & \sum ...
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58 views

Show that in $\mathbb{Z}[x,y]$, $\langle x+y,x-y\rangle\subsetneq\langle x,y\rangle$

I have shown that in $\mathbb{Q}[x,y]$, $$\langle x,y\rangle=\langle x+y,x-y\rangle,$$ by stating that $$\{x,y\}\subset\langle x+y,x-y\rangle$$ because both $x$ and $y$ can be written as elements of ...
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Constructing multivariate polynomial from roots

A polynomial that should have (real) roots at $x_1, x_2, ...$ can easily be constructed with the factored form of a (univariate) polynomial $p(x) = (x-x_1) (x-x_2) ...$. How about the multivariate ...
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Prove: A basis $G$ is a Grobner basis of an ideal $\iff$ for every element $S$ in a homogeneous basis for the syzygies $S(G)$ we have $S.G \to_{G}0. $

A basis $G=(g_1,...,g_t)$ for an ideal I is a Grobner basis $\iff$ for every element $S=(h_1,...h_t)$ in a homogeneous basis for the syzygies $S(G)$ we have $S.G = \Sigma_{i=1}^{t} h_ig_i \to_{G}0. $ ...
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65 views

Finding Curve to Minimize Polynomial over Unit Cube

You are given an $n$-variable multilinear polynomial (can assume of degree 2 if it helps) from an $n$-dimentional cube $p:[-1,1]^n\to\mathbb{R}$ and two vertices $v,u\in\{-1,1\}^n$. Our goal is to ...
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Does there exist a bivariate polynomial that is positive exclusively in the 1st quadrant?

Does there exist a bivariate polynomial $p \in \Bbb R[x,y]$ that is positive iff $x, y > 0$? My motivation was originally to state multiple positivity conditions with one expression but now I'm ...
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Determinant as a polynomial

I am trying to understand the notion of a determinant of an $n \times n$ matrix as a polynomial of degree $n$ in the entries of a matrix. If I wrote a matrix of the form $$\begin{bmatrix} a & b \\ ...
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Hyperbolic polynomials: proof that a polynomial is hyperbolic in every direction $v$ taken from its hyperbolicity cone

Let $p$ be a hyperbolic polynomial in the direction $e\in\Bbb R^n$. Then it is also hyperbolic with respect to every direction $v\in\Lambda_{++}:=\{x:p(x-te)=0\implies t>0\}$. This is from this ...
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Vector space of multivariate polynomials in Magma

I would like to define the $K$-vector space generated by a finite basis of multivariate polynomials (in $n$ variables, over the field $K$). My goal is then to find out the components of any element in ...
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$X^3Y+Y^3+X\in \Bbb Z_2[X,Y]$ is irreducible in $\Bbb Z_2[X,Y]$

Exercise. We would like to prove that the polynomial $f(X,Y):=X^3Y+Y^3+X\in \Bbb Z_2[X,Y]$ is irreducible in $\Bbb Z_2[X,Y]$. My attempt. Consider $f(X,Y):=X^3Y+Y^3+X\in (\Bbb Z_2[X])[Y]$ and we are ...
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Generalizing a property of one-variable rational functions to multiple variables. [duplicate]

I ask this question in response to a discussion on this question. If $f(x)=p(x)/q(x)$ is a one-variable real rational function, then for each $r\in \mathbb{R},$ $f(x)=r$ has at most $d=\max\{\deg(p),\...
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58 views

Degree of a multivariate polynomial over a finite field with many roots

Question Let $q$ be a prime power, $k\in\{1,\ldots,q-1\}$ and $f$ be a multivariate polynomial in $\mathbb{F}_q[x_1,\ldots,x_n]$ having $q^n - k$ roots. Show that $\deg(f) \geq (q-1)n - k + 1$. (The ...
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55 views

Get a random circle C inside a bigger circle, and C encompasses a specific point

So basically I want to draw a random circle of radius R inside a bigger one, but the drawn circle should encompass a specific point. If my math is right, it comes down to solving the following and ...
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How can I integrate over the product of two multivariate normal distributions?

Suppose that $\mathbf{y} \sim N(\mathbf{n},\sigma^2\mathbf{I})$ and $\mathbf{n} \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma})$. I want to integrate the following: $$\int [\mathbf{y}\mid\mathbf{n},\...
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Degree of spherical polynomial

How is it defined? For example in dimension 2, $P(x,y) = x^2 + y^2$ is a bivariate polynomial of algebraic degree 2, but when $(x,y)$ is a point of the unit circle we have $P(x,y) \equiv 1$ which is ...
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Clarification on a solution for Atiyah-Macdonald Chapter 1 Q.3.

The question is supposed to be a generalisation of Chapter 1 Q.2 to multivariable polynomials. However, I am specifically referring to the statement: $f$ is a zero-divisor iff there exists $a\in A$ s....
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32 views

Is this function coercive? Is my method reasonable?

Is this function coercive? $$f(x) = (x_1+2x_2)^2$$ I thought that because, for $x=(t, -t/2)'$ (whose norm goes to infinity when t does), $f(x)=0$, that this would mean that $f$ is not coercive, but ...
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Curvature of multivariate polynomial at the origin

For any Gaussian curvature $K$, the polynomial $x^2 + \frac{K}{2}y^2$ has Gaussian curvature of K at the origin. Let $M$ be a Riemann Manifold of dimension $n$ and let $x$ be a point in $M$. Does ...
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Quickly finding a solution to a single restricted polynomial function with multiple variables

I am interested in finding roots for multi-variable polynomial functions over the real numbers. I looked around and found some solutions that apply to a more general problem, but they seemed ...
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101 views

Is $R[[x]][[y]]$ the same as $R[[x,y]]?$

Let $x,y$ be two central indeterminates in $R$, is $R[[x]][[y]]=R[[x,y]]$? My take on this is: Let $f(x,y) \in R[[x]][[y]]$, then $f(x,y) = \sum_{i=0}^{\infty}g_i(x)y^i = \sum_{i=0}^{\infty}\sum_{j=0}...
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32 views

Divisibility of my special general polynomial [closed]

Let $P_0(y),P_1(y),...,P_n(y)\in\mathbb{C}[y]$ where $P_n(y)\neq 0$, and let $p(x),q(x)\in\mathbb{C}[x]$ coprime, both nonzero and $q(x)$ nonconstant. Consider the multivariate polynomial $$P(x,y)=...
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Irreducibility of polynomials of two variables over the algebraic numbers?

Let $n\in\mathbb{N}_0$, $P_0(y),...,P_n(y)\in\mathbb{C}[y]\ \ \ \ ($or $\overline{\mathbb{Q}}[y])$, $p(x),q(x)\in\mathbb{C}[x]\ \ \ \ ($or $\overline{\mathbb{Q}}[x])$, $p(x)$ and $q(x)$ coprime. Let $...
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Primeness of an ideal generated by two polynomials in 5 variables

Let $R = \mathbb{C}[x,y,z,w,t]$ be a polynomial ring in 5 variables. Consider the following polynomials (which come from the classical invariant theory of forms): $f = z^{2} - 3 y w + 12 x t - \alpha$...
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Using a partial derivative to estimate next order partial derivatives

Came across this in one of the books while calculating the Hessian of a function that has a discontinuity. Not able to understand that how is the evaluation of first-order partial derivative w.r.t to ...
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71 views

Polynomial Expression in Multivariate Normal Distribution

I am trying to work out the polynomial expression in multivariate normal distribution. The inner product inside the epsilon for univariate (assuming 0 mean and unit variances/covariances for ...
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92 views

Efficient way of getting coefficients of a generating function for a system of linear equations

I was playing trying to count how many union-closed families $\cal{F} \subseteq \cal{P}(\{a, b, c\}) \setminus \emptyset$ are there. I know there are $61$ such families (see for example this OEIS ...
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Vieta's formulas for multivariate polynomials

Is there an equivalent of Vieta's formulas, relating the polynomial coefficients with its roots, for multivariate polynomials? I tried to search for references but could not find anything.
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$Y^3+XY^2+X^3Y+X$ is irreducible in $\mathbb{C}[X,Y]$

I want to prove that the polynomial $$Y^3+XY^2+X^3Y+X$$ is irreducible in $\mathbb{C}[X,Y]$. If I try to write it as product of two polynomials $g$ and $h$, I can see that in order for $Y^3$ to appear,...
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Does a multivariate polynomial stay irreducible in transcendental field extension?

If $F$ is an algebraically closed field, and $p \in F[X_1, ..., X_n]$ irreducible, and $K$ a (transcendental) field extension of $F$, is $p$ always irreducible in $K$?
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80 views

Hilbert's Nullstellensatz for Laurent Polynomials

I am picking up abstract algebra by myself and understood Hilbert's (weak) Nullstellensatz theorem for polynomials in $\mathbb{C}[\pmb{x}]=\mathbb{C}[x_1,\dots,x_n]$. I am now trying to understand the ...
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Multivariate Polynomials of Max-degree 1

Definition [Monomial of max-degree 1]. Given $n$ variables ${x_1, ... ,x_n}$, a multivariate monomial of max-degree 1 is an expression of the form: $r(x_1^{e_1} \cdot x_2^{e_2} \cdot \dots \cdot x_n^{...
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38 views

Solve this system of equations with three variables

Solve for positive solutions: $a+b^2+2ab=9$ $b+c^2+2bc=47$ $c+a^2+2ac=16$ What I've done so far is add up all the equations to get: $a+b+c+(a+b+c)^2=72$, from which I got: $a+b+c=8$, but I still ...
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$x$ and $y$ are real numbers, $x^2-2x-4y=5$, what is the range of $x-2y$?

$x$ and $y$ are real numbers, $x^2-2x-4y=5$, what is the range of $x-2y$? I think I'm supposed to rewrite the equation in terms of $(x-2y)$ and get a quadratic, but I'm not sure if I'm really ...
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How to find polynomial multipliers that relate a specific element in the Groebner basis to the original generators?

If I have a particular set of generators for an ideal, (eg. $f_1$, $f_2$, $\ldots$, $f_n$), it is obviously straightforward to compute the Groebner basis for this ideal (say, $g_1$, $g_2$, $\ldots$, $...
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47 views

Maple simplify a polynomial without using functions that uses gcd.

Currently trying to find a way to simplify a polynomial in $x,y$ without using functions that uses gcd, eg. simplify(). And failing spectacularly. $2 + 3*x + 3*x^3*y + 3*x*y - x^4*y^3 - x^3*y^4 - x^2*...
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66 views

Analytical solution of $ax+by+cz+dw+… = n$

Suppose I have a multivariate linear equation (of an arbitrary number of variables, for this example, say 4): $f(x,y,z,w) = ax+by+cz+dw=n$, where $a,b,c,d$ are constants that I know. And $n$ is the ...
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1answer
38 views

Generating Reducible polynomials of a particular type

I am working on a paper and I need to know if there exists any REDUCIBLE bivariate polynomials of the following form: $Ax^4+2Ax^3+2Bx^2y-(2C+1)y+D=0$ Where A,B,C,D are all positive integers I ...
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191 views

How to solve this system of hyperbola equations?

I have a system of 2 equations each describing the branch of a hyperbola. The below equations represent hyperbolae with foci $P_0$ and $C_1$ (or $C_2)$ and transverse axis length $r_1$ (or $r_2)$. I'...
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24 views

Is there an algorithm to factor a many variable polynomial into a sum of squares?

I have a homogenous degree 8 polynomial in 14 variables. I know it is possible to express it as a sum of 8 squares, but it's a very complicated polynomial and it's infeasible to manipulate it by hand. ...
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32 views

Expansion of $(\sum x_i)^n$ in Schur polynomials

I have implemented the Schur polynomials in a computer, following two different methods. I was intrigued by this claim from an article: I find this paragraph a bit unclear. I'm not sure whether it ...
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1answer
53 views

fitting a multivariate polynomial to 3D data

I collected data about the response function, $f$, for each and every combination of three parameters $x$, $y$ and $\theta$, that represent horizontal position, vertical position and orientation, ...
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73 views

Condition for Projective Variety

The definition of projective variety is equivalent to the locus of zero set of homogeneous polynomials that generate a prime ideal in the algebraically closed field they are a subset of (polynomial ...
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1answer
140 views

Maximal ideal and not algebraically closed field

This is a question concerning maximal ideals in a polynomial ring over a non-algebraically closed field k. First is the example inspired for this question: as a standard exercise, it is easy to show ...