# 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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### 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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### 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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### 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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### 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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