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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How to solve system of 2 arbitrary bivariate quadratic equations over finite field?
I'm in the process of needing a solver for bivariate quadratic system of 2 equations over finite field - this is to estimate the time complexity of breaking an algorithm that I'm designing.
Most ...
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Conjugates of Algebraic Power Series
Let $\mathbb{K}$ be a field of characteristic zero. Let $\mathbb{K}[x]$, $\mathbb{K}(x)$, $\mathbb{K}[[x]]$, $\mathbb{K}((x))$ be the ring of polynomials, the field of rational function, the ring of ...
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Number of monomials in product of total-degree-bounded dense polynomials and connection to Fuss–Catalan numbers
I have a combinatorial question involving multivariate polynomials.
Let $n$ be a positive integer, and consider $n$ polynomials $p_1, p_2, \ldots, p_n$, where each $p_k$ is a dense polynomial in $k$ ...
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Evaluation of the Composite of Multivariable Power Series (Reference Request)
I am looking for a reference for the following result:
Theorem. Let $K$ be a complete valued field, $m,n\in \mathbb{Z}_{\geq 1}$, power series $$f_1,\dots,f_m\in (X_1,\dots,X_n)K[[X_1,\dots, X_n]]\...
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Unique homogeneous polynomial determined by hyperplanes
Let $H_0,\ldots, H_d\subset \mathbb{R}^n$ be vector hyperplanes such that the intersection of any $r$ of them has codimension $r$ for every $r\leq n$. For $i=0,\ldots, d$ let $g_i$ be a degree $d$ ...
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Dominating Multivariable Polynomials
Is there a direct way of proving that every multivariable polynomial is dominated by some multivariable polynomial?
In the proof of Richardson's Theorem, one talks about a dominating function for ...
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Karatsuba like algorithm for multivariate polynomial
I have a fairly big expression of a multivariate polynomial looking something like
$$
P(x_0,x_1,...,x_n) = \sum_{i=0}^n \prod_{j} x_{\text{rand}j}
$$
Sorry for the loose notation, it's a sum of ...
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Gröbner basis computations with variable modulus
I am interested in finding a solution to a system of multivariate equations with varying prime power modulus.
For example, finding solutions $(a,b,t) \in (\mathbb{F}_{1009})^3$ to the following:
$101a ...
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Convexity of multivariate polynomials in tensor form
I am considering the convexity of multivariate polynomials in the form of, say, $Px^4$, where tensor $P$ is a 4-order n-dimensional positive definite (that is, $\forall x\neq 0, Px^4>0$) symmetric ...
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Is $x^my^n+x^py^q+1$ irreducible in $\mathbb{C}[x,y]$?
I am trying to prove that the bivariate polynomial $ x^m y^n + x^p y^q+ 1 $ is irreducible in $ \mathbb{C}[x, y] $, where $ m, n, p, q \in \mathbb{Z}_+ $, $m$ is coprime to $p$ , $n$ is coprime to $q$ ...
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Valuation ("order of vanishing") on multivariable polynomial ring R[x1,...,xn] (in Roth's theorem on Diophantine Approximation)
Let $R$ be $\mathbb Z$ or a (char-$0$$\color{red}{?}$) field (I think in general, a characteristic $0$ integral domain should work$\color{red}{?}$ But perhaps someone can clarify this). In general for ...
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approximation on determinant of a matrix
I wonder if there exists a mapping $f:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}$ such that there exists two constants $c_1,c_2>0$ independent of $n$ holding
\begin{align*}
&{\bf (i)}:\qquad ...
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Show that every set of positive measure in $R^n$ must contain a basis for $R^n$
The intuition comes from the fact that every open set in $R^n$ must contain a basis for $R^n$. Now, I want to extend this to a set of positive measure. But I don't know how to start it.
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What can be said about the degree of the remainder in bivariate polynomial division?
Consider the following polynomials in 2 variables: the dividend $p(x,y)$, the (linearly independent) divisors $f_1(x,y), \dots, f_s(x,y)$, the quotients $q_1(x,y),\dots,q_s(x,y)$ and the remainder $r(...
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How to find a reccurence formula like Newton's identity for $\sum_{i<j} x_i^m x_j^m$?
In the question below the $e_i$ correspond to the elementary symmetric polynomials.
Using the Mathematica function FindLinearReccurence I found that
$$
q_m = x_1^m ...
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