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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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Proof of a method of calculating the ideal quotient $I:\langle f^{\infty}\rangle$

Background: (1)Let $I$ be an ideal of $K[x_1...x_n]$,and let $f\in K[x_1...x_n]$,where K is a field. We define $I:f^{\infty}:=\bigcup_{i=1}^{\infty}I:\langle f^i\rangle$. And I have known that $\...
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Polynomial formula for orthogonal vector in odd dimensions [duplicate]

I have been thinking about this problem recently. In 2 dimensions there is an easy formula for a nonzero vector orthogonal to a given vector $(x, y)$, namely $(-y, x)$. By taking pairs of coordinates, ...
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How to programmatically perform multivariate polynomial division modulo 2.

I'm trying to write code to do multivariate polynomial division modulo-2 (so no coefficients or degrees). Say I have the polynomial $xyzw + yz$ and want to divide it by $xw + xwz$. IIUC, the usual ...
Aaron's user avatar
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Leading term of difference of two products

$n$ is a positive integer. Let $$P_n(x)=\prod_{\substack{\mu_1,\dots,\mu_n\in\{\pm1\}\\\mu_1\dots\mu_n=1}}(x-(\mu_1{x_1}+\dots+\mu_n{x_n}))\\ Q_n(x)=\prod_{\substack{\mu_1,\dots,\mu_n\in\{\pm1\}\\\...
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Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?

Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$ where $r>...
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Finding the real roots of a set of multivariate polynomial in an interval

Problem: I have a set of $m$ multivariate polynomials over $k$ variables, with an upper bound on the degrees $n$: $$ \{f_i(x_1,x_2...x_k)\}_{i=1}^{m} $$ My goal is to find if there's a real root such ...
Waylander's user avatar
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How to invert system of multivariable polynomial equations for dependent variables

I was wondering if I could get some input on my strategy for inverting a set of multivariate polynomial equations for the independent variables. My functions are as follows: $$g_1(x,y) = a_0 + a_{1}x+...
Mr Phase Locked Loop's user avatar
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Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
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Existence of constrained pairs of elements in $\mathbb{C}[x,y,z]/(x^3,y^3,z^3)$ that multiply to zero

Question: Do there exist a pair of elements $a,b\in \mathbb{C}[x,y,z]/(x^3,y^3,z^3)$, where $a\notin(x,y)\cup(y,z)$ and $b\notin (x,z)\cup (y,z)$ but $ab=0$? For context, I'm trying to show that for a ...
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Is it possible to generalise the values of $a$ and $b$ for these following expressions?

$a{}^b$ = m $b{}^a$ = n Find $a$ and $b$ in terms of $m$ and $n$ Such that $a,b,m,n$ $\in {}^+R$ I tried logging both sides and substituting the values. It forms an equation $n{}^\frac{1}{a} \ln (a) - ...
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Is it possible to know or get an reasonable upper bound on the number of solutions to an $2$D Polynomial equation ($n$D)?

I have two arbitrary but finite order $2$D polynomial functions $P_1(\vec x)$ and $P_2(\vec x)$, both on the domain $[-1,1]^2$. If I construct an algebraic equation $P_1(\vec x)^2 + P_2(\vec x)^2 = 0$,...
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Why can't individual terms of a summation not cancel each other in the 2nd case?

Below is from a paper. $F(.)$ is a low-degree multivariate polynomial over $\mathbb F$ in $s$ variables. Checking if $\sum_{x \in \lbrace0,1\rbrace^s} F(x) = 0$ will not prove that that $F(x) = 0\...
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Find a multivariate polynomial over finite field with given zeros (or number of zeros) with an upper bound on its degree

This is a follow-up on: Find a multivariate polynomial over finite field with given zeros (or number of zeros). I am trying to find a polynomial $f \in \mathbb{F}_q[x_1, x_2, \dots, x_m]/(x_i^q-x_i)$ ...
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Multivariable Function Approximation Review Paper

I am doing some research on function approximation for a general multivariable equation. (i.e., I have a function f that depends on $x_1, x_2, x_3, ..., x_n$. Given many data points of f at multiple ...
Thian Daniel Iskandar's user avatar
4 votes
1 answer
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Find a multivariate polynomial over finite field with given zeros (or number of zeros)

I am trying to find polynomials $f \in \mathbb{F}_{q} [x_1, x_2, \dots, x_m]/(x_i^p-x_i)$ such that $f-1=0$ has precisely a given number of roots. For example, $f(x, y)$ in $\mathbb{F}_5[x, y]/((x_1^5-...
Tanay Saha's user avatar
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How to construct Legendre polynomials for $x_1,...,x_k$?

I'm trying to run a nonparametric regression to estimate the unknown conditional mean $E(Y|X_1=x^*_1,X_2=x^*_2)$ using data set $\{Y_i,X_{1i},X_{2i}\}_{i=1}^n$. This could be done by nonparametric ...
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Multivariate Taylor Theorem Peano remainder

I'm a bit lost on this problem. I want to show that if a function $f\colon U\subset \mathbb R^n\to \mathbb R$ is of class $C^N$ on $U$ (continuous partial derivatives up to order $N$) and $T_N(\hat{x})...
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Show any root of an elementary symmetric polynomial does not have positive imaginary part in all it's components

We define $H=\{u \in \mathbb{C}: \text{im}(u)>0\}$ to be the open upper half plane. For $n \in \mathbb{N}$ and $k = 1, \ldots n$ let $e_{n,k}(x_1,\ldots,x_n)=\sum\limits_{1 \leq j_1 < \ldots <...
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How many evaluations do you need to prove that a multivariate polynomial is the zero polynomial?

For a univariate polynomial $f$, you just need to prove that $f(x) = 0$ for $d+1$ distinct $x$ to prove that $f$ is the zero polynomial. But for multivariate polynomials, how does that work? How many ...
poopipoo's user avatar
2 votes
2 answers
173 views

Algorithm to simplify division of two multivariate polynomials

I'm looking for al algorithm to try to implement simplification of a fraction of two multivariate polynomials. Here's an example of one: $$\frac{(3xy^2+x^2y^2+7yx^2+21xy+2y^2+12x^2+14y+36x+24) }{(4xy^...
floomp's user avatar
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3 votes
1 answer
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split linear factor from multivariate polynomial

This is a bit embarrassing (and hopefully hasn't been asked before). Let $F$ be a field and $p(x,y)$ be a polynomial in $F[x,y]$. Suppose that $p(x,x)=0$. For most authors it is obvious that this ...
Brauer Suzuki's user avatar
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Partial fraction decomposition for multivariate rational functions

Is there an analogue of partial fractions for multivariate rational functions? For example, what would be a good way to express $f(x,y) = \frac{1}{(x+y)(x-y)}$?
Chris Jones's user avatar
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Characterizing a complex multivariate polynomial by its zeroset

Let $\mathbb{C}[X_1,...,X_d]$ denote the set of polynomials over $\mathbb{C}$ in $d$ variables. Let $\mathcal{A}$ denote the set of all $z \in \mathbb{C}^d$ where there exists $1 \leq i \leq d$ such ...
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Is there a general formula for factorizing multivariate complex polynomials?

Let $\mathbb{C}[X_1,...,X_d]$ denote the space of polynomials over $\mathbb{C}$ in $d$ variables. In the case $d = 1$, we can always factorize a polynomial using its zeros and their multiplicity, i.e. ...
Andreas132's user avatar
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Characterization of irreducible multivariate polynomials over the complex numbers

Let $\mathbb{C}[X_1,...,X_d]$ denote the set of polynomials over $\mathbb{C}$ in $d$ variables. If $d=1$, we know that the irreducible polynomials are exactly the polynomials of degree $1$, i.e. ...
Andreas132's user avatar
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Homogeneous quadratic parts of polynomials in Unbalanced Oil and Vinegar

In the book Multivariate Public Key Cryptography, the author describes the polynomials in the cryptographic system Unbalanced Oil and Vinegar in the following way: Define $V=\{1,\dots,v\}$ and $O=\{v+...
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Criterion for unicity and existence of pre-image in multivariate cryptography

I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called bipolar construction where one chooses three maps to construct encryption and signature ...
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1 answer
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Factoring $x^2(y+z) + y^2(x+z) + z^2(x+y)$

I have the polynomial $$x^2(y+z) + y^2(x+z) + z^2(x+y)$$ that I want to factor. I know how to factor $x^2(y-z) + y^2(z-x) + z^2(x-y)$. Because $(y - z) + (z - x) + (x - y) = 0$, we can use the ...
py_math's user avatar
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1 answer
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How to solve this system of multivariate polynomial equations for $0<x_7<x_6<x_8 \le 1$? Groebner basis maybe?

I am reformulating my question according to the guidelines I was given. I have the following problem: I cannot find a way to solve the system of equations further down. This is the calculations from ...
fabs's user avatar
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Conditions for overdetermined system of homogeneous polynomials admit unique solution

I have a system of complex polynomials equal to zero with $n$ variables and each polynomials is homogeneous in degree. There are $m$ polynomials and $m>n$. So the system is overdetermined. Since ...
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How to find roots of a system of multivariate polynomials? [closed]

I am trying to find the roots of a system of 3 multivariate polynomials with 3 variables. The polynomials are really 'ugly'. So far I have tried to find a Groebner Basis in Maple and got a Groebner ...
fabs's user avatar
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1 vote
1 answer
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Why homogeneity of an equation is preserved even when we change variables?

Consider the equations, $$x^{2}+y^{2}+z^{2}-xt-t^{2}=0 \tag{1}$$ $$x^{2}+y^{2}+z^{2}+yt-2t^{2}=0 \tag{2}$$ Clearly, both equations are homogeneous. Solve for $t$ from the above equations. You will get ...
Sasikuttan's user avatar
1 vote
2 answers
638 views

How many roots does a polynomial in 2 variables have?

We know that in general a polynomial in one variable of degree n has n roots. What about in two variables? It seems instead of discrete points we have whole regions. For example: $xy = 0$ has as ...
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How to express cubic (and higher) forms in matrix-vector notation?

In linear algebra we often learn about linear functions $$f_1(x) = Ax+b$$ And Quadratic forms ; $$f_2(x) = x^tAx + Bx + c$$ However we could easily imagine arbitrary high level polynomials: $$\sum c_{...
mathreadler's user avatar
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2 votes
1 answer
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Double zeros of a two variable polynomial encoded in a complex function

Say I have a two variable polynomial $$f(x,y)=xy(1-x-y)$$ where $x$ and $y$ are real. The solutions to $f(x,y)=0$ are the three lines $x=0$, $y=0$ and $x=1-y$. I am interested in the "double ...
Giulio Crisanti's user avatar
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How to check if $f(X) \in F(g_1(X), \ldots, g_m(X))$?

Let $X=(X_1,\ldots, X_n)$ and $g_1(X), \ldots, g_m(X) \in F(X)$ for a field $F$. Which algorithms or methods are used to determine if $f(X) \in F(g_1(X), \ldots, g_m(X))$ is true for a given $f(X) \in ...
Valentin's user avatar
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1 answer
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Simplifying and Manipulating Summation

Let $\tau=(\tau_1,\tau_2,\ldots)$ be a sequence of nonnegative integers which are finitely supported, i.e. only finitely many terms are nonzero. Let $$\rho=(\rho_j)_{j\geq 1}\in\mathbb{R}^{\mathbb{N}}....
Nicolas's user avatar
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Factorization of multivariate polynomials

I am trying to factor a polynomial over $\mathbb{C}$ of degree 4 in 4 indeterminates, but I realized I don't know any algorithm to do so and no software to help me. Is there an algorithm to factor ...
MRAA's user avatar
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How can I sort the values of variables of multivariate polynomial f by increasing order of value of f

Assume that I have a multivariate polynomial over positive integers, the coefficients $a_i$ are all 1, e.g . $f(x_1,x_2,x_3,x_4)=x_3^4+x_2+x_1^2x_2x_4^5$, each monomial and each $x_i > 0$. I have ...
Sefi Potashnik's user avatar
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Maple - Express Multivariate Polynomials in Specific Ideals

I want to know if there exists a command in Maple that can show how a multivariate polynomial is expressed in the specific ideal, and show what this expression is. ex: $$D4 := a_0^3*a_4^3 - 12*a_0^2*...
Zero's user avatar
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1 answer
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What are the conditions on two multivariate real polynomials $f$ and $g$ so that their zero sets are similar?

Basically what the title says. Let $f$ and $g$ be two polynomials with real coefficients of degree $m$, in $n$ real variables. The solutions of the equation $f(x_1,\ldots,x_n) = 0$ then define a ...
paulina's user avatar
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1 vote
1 answer
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Is a set defined using a polynomial identity an algebraic variety

We consider a polynomial $F \in \mathbb{C}[X_1, \ldots , X_n, Y_1, \ldots , Y_m]$ in $n+m$ variables. Define the set $A_F = \{ x \in \mathbb{C}^n \mid F(x_1, \ldots , x_n, Y_1, \ldots , Y_m) = 0\}$ ...
leon.fuchsler's user avatar
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1 answer
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Enumerating solutions of incomplete row reduced echelon form over multivariate polynomial ring over GF2

I'm currently experimenting with finding systems of multivariate equations, matching a given set of inputs with their roots. I'll explain with a toy example: Given four "valid" assignments ...
DasArchive's user avatar
13 votes
0 answers
330 views

On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS. Definition 2. If an $n$-variate polynomial ...
khashayar's user avatar
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1 vote
2 answers
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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 $\sum_{i=0}^n f_iY^i$, ...
Gregori 's user avatar
1 vote
3 answers
99 views

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 ...
user1145925's user avatar
2 votes
0 answers
93 views

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 (...
Hidda Walid's user avatar
1 vote
0 answers
27 views

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 ...
Stéphane Laurent's user avatar
3 votes
0 answers
55 views

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 ...
Zach H's user avatar
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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, ...
Jacques C.'s user avatar