# 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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### 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 ...
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### Hyperbolic polynomials: proof that a polynomial is hyperbolic in every direction $v$ taken from its hyperbolicity cone

Let $p$ be a hyperbolic polynomial in the direction $e\in\Bbb R^n$. Then it is also hyperbolic with respect to every direction $v\in\Lambda_{++}:=\{x:p(x-te)=0\implies t>0\}$. This is from this ...
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### Vector space of multivariate polynomials in Magma

I would like to define the $K$-vector space generated by a finite basis of multivariate polynomials (in $n$ variables, over the field $K$). My goal is then to find out the components of any element in ...
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### $X^3Y+Y^3+X\in \Bbb Z_2[X,Y]$ is irreducible in $\Bbb Z_2[X,Y]$

Exercise. We would like to prove that the polynomial $f(X,Y):=X^3Y+Y^3+X\in \Bbb Z_2[X,Y]$ is irreducible in $\Bbb Z_2[X,Y]$. My attempt. Consider $f(X,Y):=X^3Y+Y^3+X\in (\Bbb Z_2[X])[Y]$ and we are ...
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### Primeness of an ideal generated by two polynomials in 5 variables

Let $R = \mathbb{C}[x,y,z,w,t]$ be a polynomial ring in 5 variables. Consider the following polynomials (which come from the classical invariant theory of forms): $f = z^{2} - 3 y w + 12 x t - \alpha$...
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### Using a partial derivative to estimate next order partial derivatives

Came across this in one of the books while calculating the Hessian of a function that has a discontinuity. Not able to understand that how is the evaluation of first-order partial derivative w.r.t to ...
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### Polynomial Expression in Multivariate Normal Distribution

I am trying to work out the polynomial expression in multivariate normal distribution. The inner product inside the epsilon for univariate (assuming 0 mean and unit variances/covariances for ...
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### Efficient way of getting coefficients of a generating function for a system of linear equations

I was playing trying to count how many union-closed families $\cal{F} \subseteq \cal{P}(\{a, b, c\}) \setminus \emptyset$ are there. I know there are $61$ such families (see for example this OEIS ...
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### Vieta's formulas for multivariate polynomials

Is there an equivalent of Vieta's formulas, relating the polynomial coefficients with its roots, for multivariate polynomials? I tried to search for references but could not find anything.
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### $Y^3+XY^2+X^3Y+X$ is irreducible in $\mathbb{C}[X,Y]$

I want to prove that the polynomial $$Y^3+XY^2+X^3Y+X$$ is irreducible in $\mathbb{C}[X,Y]$. If I try to write it as product of two polynomials $g$ and $h$, I can see that in order for $Y^3$ to appear,...
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### Does a multivariate polynomial stay irreducible in transcendental field extension?

If $F$ is an algebraically closed field, and $p \in F[X_1, ..., X_n]$ irreducible, and $K$ a (transcendental) field extension of $F$, is $p$ always irreducible in $K$?
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### Hilbert's Nullstellensatz for Laurent Polynomials

I am picking up abstract algebra by myself and understood Hilbert's (weak) Nullstellensatz theorem for polynomials in $\mathbb{C}[\pmb{x}]=\mathbb{C}[x_1,\dots,x_n]$. I am now trying to understand the ...
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