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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Polynomial and rational functions of sets

I'm trying to understand Blass 1984's proof that, in $\mathsf{WZF}$, all vector spaces having a basis implies $\mathsf{AMC}$. Given a field $k$ and pairwise disjoint nonempty sets $X_i$ of union $X$, $...
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Show that $g$ can be written as $g=(X^2-Y)q+r$ for $q \in \Bbb R[X,Y]$ and $r \in \Bbb R[X]$.

Let $p=X^2-Y \in \Bbb R[X,Y]$ and let $g \in \Bbb R[X,Y]$. Show that $g$ can be written as $g=pq+r$ for $q \in \Bbb R[X,Y]$ and $r \in \Bbb R[X]$. Hint: $g$ can be written as $\sum_{i=0}^n f_iY^i$, ...
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Find solution to multivariate equation in the infinite limit for one variable.

I have the below equation (where $\{x|x\in\mathbb{R},0\lt x\lt \frac{1}{2}\}$ and $\{n|n\in\mathbb{N},n\geq5\}$): $-(\frac{x}{1-x})^{n-1}=\frac{-x(n+2)+2}{-x(n+2)+n}$ A) Ideally, I want to determine a ...
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Mapping square to a quadrilateral

In 1 dimensional space, to map an interval $[0,1]$ into another interval $[a,b]$ we use the function $$T: [0,1] \to [a,b] :\quad x \mapsto (1-x)a+xb$$ What is the generalisation of this in 2D space (...
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Zonal quaternionic polynomials

I've done a R package which computes the Jack polynomials (I started it a couple of years ago). It also computes the zonal polynomials (Jack with $\alpha=2$ up to a normalizing factor), the Schur ...
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Polynomials evaluated on all subsets

Let $f(x_1,\dots,x_k)$ be a multivariate polynomial of degree $m$. It is a seemingly well known fact that $$ g(n) = \sum_{1 \leq i_1 < \dots < i_k \leq n} f(i_1,\dots,i_k) $$ is a polynomial of ...
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Continuity Concept of Multivariable Functions

I have recently started going into limits and continuity of multivariable functions and I was just wondering if there is an easy way to determine the continuity of various functions (e.g. polynomial, ...
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The value of $f_x((0,0))+f_y((0,0))+f_{xy}((0,0)) $ is?

Let $f:\mathbb{R}^2 \Rightarrow \mathbb{R} $ defined by $$f((x,y)) = x^2 +2y^2 -3xy, \forall (x,y) \in \mathbb{R}^2$$ The value of $f_x((0,0))+f_y((0,0))+f_{xy}((0,0))$ is: $0$ Undefined $-3$ 3 I'm ...
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Solving systems of multivariate equations

I am new to the topic and I am working with the multivariate systems of equations over finite fields. My goal is to solve them. I know that in general the problem of solving such systems is NP-hard. ...
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Factorizing a quartic expression to show that it is a perfect square. [duplicate]

Show that $\frac{a^4+b^4+(a+b)^4}{2}$ is a perfect square. I tried this, $$\frac{a^4+b^4+(a+b)^4}{2}$$ $$\frac{a^4+b^4+(a^2+b^2+2ab)^2}{2}$$ $$\frac{2a^4+2b^4+4a^2b^2+2(a^2b^2+2a^3b+2ab^3)}{2}$$ $$a^...
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Uniqueness of affine forms

Lets define the power symmetric group $$\text{psg}_{n,d}(x_1,\dots, x_n) = \sum_{i=1}^n x_i^d$$ And lets define a linear form as $$\ell_i(x_1,\dots,x_n)=\sum_{j=1}^{n} a_{ij}x_i+a_{i0}$$ Where $a_{ij} ...
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Factorize $x(y^2 - z^2) + y(z^2 - x^2) + z(x^2 - y^2)$ [closed]

I know the answer is $-(y-z)(x-y)(x-z) = (x-y)(y-z)(z-x)$ via WolframAlpha. However, I don't why that's the solution nor how to get it "by hand". (nor what specific area of math I can find ...
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Show that a sequence of polynomials cannot be a Gröbner basis wrt. any term ordering

Consider the ideal $I=\left\langle x^{2}+y^{2}, x^{3}+y^{3}\right\rangle \subseteq \mathbb{Q}[X, Y]$ and the basis $G=(x^2+y^2,x^3+y^3)$ with lexicographic ordering $x\geq y$. Consider two $S$-...
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How to know whether the off diagonal entries are positive or negative? What is the relationship between off diagonal entries and covariance?

enter image description here According to what I learned, off diagonal entries are Cov(x,y) and Cov(y,x).
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alternative representation of vector valued taylor approximation

Im having trouble solving in between two representations of a taylor approximation. The standard (non summation) notation for taylor approximations of vector valued functions as per Magnus and ...
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Kernel of ring homomorphism in polynomial ring

Let $K$ be a field with $a_1,...,a_n \in K $ and $\phi : K[x_1,...,x_n] \rightarrow K , \ \phi (f)=f(a_1,...,a_n) .$ I am trying to show that the kernel of this ring homomorphism is the ideal $ I:= (...
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Property of the symmetric reduction of a multivariate polynomial

The following symmetric multivariate polynomial: $$P(x_1,\ldots,x_n)=\prod_{1 \le i \lt j \le n}{\left(1+x_{i}x_{j}\right)}$$ has the following property: the number of classes of its monomials, ...
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Algorithms for expressing symmetric polynomials as a polynomial in the elementary symmetric polynomials

Due to the fundamental theorem of symmetric polynomials every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials (here an example). I know from here and here ...
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Finding multivariate polynomial coefficients from values at points.

I found an answer for how to determine the coefficients of a polynomial $P(x)$ with all nonnegative integer coefficients, knowing its value on one or two points. How could that method be extended to a ...
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Using 3D Piecewise Functions to Model a Rollercoaster

I am designing a roller coaster using functions (ie. linear, cubic, logarithmic, trigonometric). At some point, one of the parts of the rollercoaster does not follow a two dimensional graph, but ...
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Count different monomials up to a permutation of variables in a generating function

Is there a clever way to count the number of different monomials, up to a permutation of the variables, of the polynomial: $$g(z_1,\ldots,z_n)=\prod_{1 \le i_1 \lt \ldots \lt i_{n-2} \le n}{\left(1+z_{...
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Factorization and roots of a multivariate polynomial over finite fields

I am interested to know whether factorization of a multivariate polynomial $f(x_1, x_2, \dots, x_n) \in \mathbb{F}_p[x_1, x_2, \dots, x_n]$ into irreducible factors yields some information about the ...
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Approximately-Global Nonlinear Multivariate High-Dimensional Optimization of *Differentiable* Nonconvex Scalar Function

Title says it all. What methods do you recommend to solve this specific problem? I can build most solutions you may suggest in Python, where I'd like the solver to be. I've spent hours and hours ...
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4 votes
1 answer
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Factorizing multivariate polynomials in $GF_2[x_1, ..., x_n]/[x_1+x_1^2,...]$

I'm trying to factorize multivariate polynomials in the quotient ring $GF_2[x_1, ..., x_n]/[x_1+x_1^2,...,x_n+x_n^2]$. For example $$ \begin{align} f&=(x_1 + x_2 + x_3)(x_2 + x_4)(x_1 + x_4) \\ &...
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Divisors of a "symmetric" polynomial

Let $k$ be a field. Let $F$ be a polynomial in $k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ with the following "symmetric" property (S): If we write $F$ explicitly as $\displaystyle F=\sum_{i} ...
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When do the kernels of surjective morphisms $\mathbb{R}[x_1,\dots,x_n]/f\mathbb{R}[x_1,\dots,x_n]\rightarrow\mathbb{R}$ have trivial intersection?

Let $\mathcal{F}$ be an $\mathbb{R}$-algebra. We write $|\mathcal{F}|$ for the set of surjective algebra homomorphisms $\mathcal{F}\rightarrow\mathbb{R}$, and we say $\mathcal{F}$ is geometric when $\...
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Is it possible to approximate multivariate functions using Chebyshev polynomials?

I have a 2D unstructured mesh with pressure defined at each point. I want to compute the local gradient of pressure (using 10-20 neighbouring points) in 2D space using better-conditioned polynomials ...
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Are multivariate polynomials determined by their values on a lattice?

In one variable, a polynomial (of any degree) is determined by its values on a finite set of points. More specifically if $p$ is a polynomial of degree $k$, and $x_0 , \dots x_{k}$ are points for ...
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Are elements in each stage of the Lasserre hierarchy convex?

The Lasserre hierarchy is a schema for proving multivariate polynomials positive via a sum of squares decomposition. At the first level, a polynomial $p$ is written $$p = \sum_i f_i^2$$ where each $\...
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How do I take the derivative of $\text{trace}(XX^TY^TYXX^T)$ with respect to the matrix $X$?

Given the matrices $Y$ and $X$, I am trying to compute the derivative of the function $f(X,Y) = \text{trace}(XX^TY^TYXX^T)$ with respect to $X\in\mathbb{R}^{d\times k}$. I am aware of the matrix ...
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Question 1.25 of Fulton's algebraic geometry: decomposing $V (Y^4 - X^2,Y^4- X^2Y^2 +XY^2-X^3)$ into irreducibles

I have been struggling with this question 1.25 of Fulton because I cannot find an example of someone doing this explicitly. I am asked to decompose: $V (Y^4 - X^2,Y^4- X^2Y^2 +XY^2-X^3)$ into ...
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Question 1.17 from Fulton's Algebraic Curves

I have been reading Fulton's Algebraic curves, chapter 1.3. The chapter starts by proving some properties about the ideal of a set of points for which the polynomials in that ideal vanish. This is ...
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(Not homework) Factoring a multivariate polynomial

I'm trying to come up with a formula to count the number of squares in an $x_1 \times x_2 \times x_3$ grid... and eventually I gave up on figuring out if the points were in a plane and threw a ...
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Combining two 2D equations to make one 3D equation

I have two variables, x and y, that I would like to make a 3D equation/graph out of. Both variables are a part of their own parabolic equation: $z=-\frac{1}{4}x^{2}+\frac{7}{4}x$ and $z=-\frac{1}{4}y^...
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2 votes
0 answers
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Algorithm for Dirac-like "matrix factorization" of polynomials

In this Numberphile video they discuss Dirac changing to matrix equations to get something like a "square root" of $t^2 + x^2 + y^2 + x^2$. In particular they discuss the following ...
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numerically solving a system of multivariate polynomials

I have a system of polynomial equations which I need to solve numerically. The one Im currently interested in has around 20 variables and a similar number of equations. All coefficients are rational ...
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Correct application of eisenstein's criterion?

I want to show that the polynomial $P(x,y,z) = z^3+3x^2y+3xy^2+3x^2z+3xz^2+3y^2z+3yz^2+6xyz$ is irreducible. I know that I can consider this as a polynomial in one variable z with coefficients in $\...
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How do I compute the minimum or maximum LINE of a bivariate quadratic function?

Given a bivariate quadratic function: $$ y = g(x, w) = a x^2 + b x w + c w^2 + d x + e w + f $$ I think I know that if $b^2 - 4 a c < 0$, then it has a single point minimum (if $a > 0$) or ...
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Ring homomorphism from $\mathbb{F}[x,y]$ to $\mathbb{F}[u]$

Is it possible to map from $\mathbb{F}[x,y]$ to $\mathbb{F}[u]$ where each monomial $x^ay^b = u^{aw+b}$ for appropriate choice of $w$? In fact this mapping $\phi(x^ay^b) = u^{aw+b}$ seems to be ...
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Can any polynomial in $\Bbb R [x]$ be written as the difference of two sum-of-squares polynomials in $\Bbb R [x]$?

Let $f \in \Bbb R[x_1,\dots,x_n]$ be an arbitrary polynomial, not necessarily non-negative. Are there always two sum-of-squares (SOS) polynomials $g, h \in \Bbb R[x_1,\dots,x_n]$ such that $f=g-h$? If ...
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Is there some special algorithm to minimize polynomial functions? [closed]

I have an optimization problem of the form $$\begin{array}{ll} \underset{\textbf{x} \in \Bbb R^4}{\text{minimize}} & f(\textbf{x})\\ \text{subject to} & \textbf{x} \in \Omega\end{array}$$ ...
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Choi Lam homogeneous polynomials as sums of squares

I came across two polynomials that Choi and Lam gave in 1976, that are not sum of squares of polynomials, despite being evidently non-negative by AM-GM $$ S(x,y,z) = x^4 y^2 + y^4 z^2 + z^4 x^2 - 3 x^...
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Factoring out $4x^3+2x^2y-2xy^2-y^3$

This can be factored as follows: $$4x^3+2x^2y-2xy^2-y^3 = (2x^2-y^2)(2x+y)$$ What is a systematic way for finding this factorization?
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Finding the analytic root of a multivariate function that involves trigonometric functions

I am currently stuck in a problem, where I need to find an analytic expression $x_0=f(\alpha,\beta)$. Let me first try to explain what $x_0$ is: Let $x$ be a real-valued variable, which must obey the ...
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The best methods for multivariate polynomial equations over finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
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Method of Frobenius Confusion

$x^2 y'' + 3xy' + (1 - 2x)y = 0$ I plugged in the guess $y = x^s \sum_{n = 0}^\infty a_n x^n$ to get $(s^2 + 2s + 1)a_0 x^s + \sum_{n = 0}^\infty (n^2 + 2ns + s^2 + 2n + 2s + 1)a_n x^{n + s} - 2a_{n - ...
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Show $ (2 - b^2c^2) (1 + b^2) \ge (b^2 + c^2)(b + c - 3)^2 $ in some range.

Show $ F = (2 - b^2c^2) (1 + b^2) - (b^2 + c^2)(b + c - 3)^2 \ge 0$ in the following two ranges of $(b,c)$: $b \in [0 \quad 1]$ and $0 \le c\le b$, $b \in [1 \quad 1.2]$ and $0 \le c\le 3-2b$. ...
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3 votes
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Finding last coefficient of a multivariate polynomial in $GF(2)$ in polynomial time

This problem was simplified by Alex Ravsky (thanks!). For the original problem see below. Simplified Problem With $f(k_1,k_2,\ldots,k_n)$ a polynomial over $GF(2)$, find the value of the coefficient ...
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When is a multivariate cubic polynomial mapping $\Bbb{R}^n$ to $\Bbb{R}^n$ both one-to-one and onto?

In one dimension, a cubic polynomial mapping $\Bbb{R}$ to $\Bbb{R}$ $$y = A + Bx + Cx^2 + Dx^3$$ is one-to-one and onto when its derivative $y'(x) = B + 2Cx + 3Dx^2$ has less than two zeros, i.e., $4C^...
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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 this ...
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