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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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 ...
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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-...
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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 ...
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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:
$$\frac{(3xy^2+x^2y^2+7yx^2+21xy+2y^2+12x^2+14y+36x+24) }{(4xy^...
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split linear factor from multivariate polynomial
This is a bit embarrassing (and hopefully hasn't been asked before). Let $F$ be a field and $p(x,y)$ be a polynomial in $F[x,y]$. Suppose that $p(x,x)=0$. For most authors it is obvious that this ...
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Factorising a multivariate polynomial in terms of products of linear polynomials using coordinate transformations
I am considering multivariate polynomials of the form
$$f(x,y)=x^a\,y^b\,p(x,y)^c$$
(and similarly for higher dimensions). I am trying to transform these polynomials into the generic form
$$\widetilde{...
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Unique Solution Polynomials
Suppose that $P_k(x,y)$ is a symmetric polynomial with real coefficients (let's say for now that it is of degree 4, but I am curious about results for general degrees). In other words, suppose that $$...
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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)}$?
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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. ...
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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. ...
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Efficiently finding a solution to the rainbow cryptosystem with quotient spaces
In this paper, Ward Beullens gives another way to look at the Rainbow cryptosystem. In it, he describes the Rainbow map $\mathcal{P}:\mathbb{F}_q^n\to\mathbb{F}_q^m$ with two layers in the following ...
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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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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)$$
that I want to factor.
I know how to factor $x^2(y-z) + y^2(z-x) + z^2(x-y)$. Because $(y - z) + (z - x) + (x - y) = 0$, we can use the ...
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How to solve this system of multivariate polynomial equations for $0<x_7<x_6<x_8 \le 1$? Groebner basis maybe?
I am reformulating my question according to the guidelines I was given.
I have the following problem: I cannot find a way to solve the system of equations further down. This is the calculations from ...
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Conditions for overdetermined system of homogeneous polynomials admit unique solution
I have a system of complex polynomials equal to zero with $n$ variables and each polynomials is homogeneous in degree. There are $m$ polynomials and $m>n$. So the system is overdetermined. Since ...
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How to find roots of a system of multivariate polynomials? [closed]
I am trying to find the roots of a system of 3 multivariate polynomials with 3 variables. The polynomials are really 'ugly'. So far I have tried to find a Groebner Basis in Maple and got a Groebner ...
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Why homogeneity of an equation is preserved even when we change variables?
Consider the equations,
$$x^{2}+y^{2}+z^{2}-xt-t^{2}=0 \tag{1}$$
$$x^{2}+y^{2}+z^{2}+yt-2t^{2}=0 \tag{2}$$
Clearly, both equations are homogeneous. Solve for $t$ from the above equations. You will get ...
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Maximization of a non-linear multi-variable symbolic piecewise function
I use math in building economic models and modeling/solving business problems.
I have a two-variable symbolic piece-wise function that I need to maximize i.e., fully and analytically characterize $(p^*...
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How many roots does a polynomial in 2 variables have?
We know that in general a polynomial in one variable of degree n has n roots. What about in two variables?
It seems instead of discrete points we have whole regions. For example:
$xy = 0$ has as ...
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How to express cubic (and higher) forms in matrix-vector notation?
In linear algebra we often learn about linear functions
$$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_{...
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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 ...
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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 ...
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Efficiently applying multinomial theorem to a large sum of terms (multivariate polynomials)
I have searched several StackExchange and StackOverflow questions and answers and have not been able to find a good solution to my problem.
Ultimately, I need to be able to efficiently find the sum of ...
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Mean and covariance of polynomial transformation of multivariate centered gaussian
Let $X\sim N(0,\Sigma)$ a centered multivariate normal distribution of dimension $n$. Let $Y=(X_1,X_2,...,X_{m<n})$ and $Z=(X_{m+1},...,X_n)$.
I am interested in computing mean and covariance of $U=...
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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 $$\rho=(\rho_j)_{j\geq 1}\in\mathbb{R}^{\mathbb{N}}....
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Factorization of multivariate polynomials
I am trying to factor a polynomial over $\mathbb{C}$ of degree 4 in 4 indeterminates, but I realized I don't know any algorithm to do so and no software to help me.
Is there an algorithm to factor ...
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How can I sort the values of variables of multivariate polynomial f by increasing order of value of f
Assume that I have a multivariate polynomial over positive integers, the coefficients $a_i$ are all 1, e.g . $f(x_1,x_2,x_3,x_4)=x_3^4+x_2+x_1^2x_2x_4^5$, each monomial and each $x_i > 0$.
I have ...
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Maple - Express Multivariate Polynomials in Specific Ideals
I want to know if there exists a command in Maple that can show how a multivariate polynomial is expressed in the specific ideal, and show what this expression is.
ex:
$$D4 := a_0^3*a_4^3 - 12*a_0^2*...
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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 ...
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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 $A_F = \{ x \in \mathbb{C}^n \mid F(x_1, \ldots , x_n, Y_1, \ldots , Y_m) = 0\}$ ...
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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 ...
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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 ...
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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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