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Let $$p_1(x_1,x_2,\dots,x_n),p_2(x_1,x_2,\dots,x_n),\dots,p_n(x_1,x_2,\dots,x_n)$$ be polynomials with $n$ variables. I would like to to find out if $$p_1(x_1,x_2,\dots,x_n)=0\\ p_2(x_1,x_2,\dots,x_n)=0\\ \vdots\\ p_n(x_1,x_2,\dots,x_n)=0 $$

have finite number of solutions?

Is there something like Bézout's theorem for this problem? I know that if $n=2$, it means that I have only $2$ polynomials of $2$ variables $p_1(x_1,x_2)$ and $p_2(x_1,x_2)$ which are coprime and homogeneous, where $deg(p_1)=m$ and $deg(p_2)=n$ then number of solutions of $$p_1(x_1,x_2)=0\\ p_2(x_1,x_2)=0\\ $$ is at least $m\cdot n$.

Are there something general which can be use for my problem? It is enough to say that if polynomials $$p_1(x_1,x_2,\dots,x_n),p_2(x_1,x_2,\dots,x_n),\dots,p_n(x_1,x_2,\dots,x_n)$$ are coprime and homogeneous, that number of solutions of equations $$p_1(x_1,x_2,\dots,x_n)=0\\ p_2(x_1,x_2,\dots,x_n)=0\\ \vdots\\ p_n(x_1,x_2,\dots,x_n)=0 $$ is finite.

Any help will be appreciated. Thank you very much.

Bernard
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Waney
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  • Where are you solving your polynomials? Over an arbitrary ring, an arbitrary field, an algebraically closed field...? – KReiser Mar 16 '21 at 22:59
  • @KReiser , I am solving the system of polynomials over the $\mathbb{R}$. If I can say something about number of solutions in $\mathbb{R}$ with some additional assumpatation to my polynomials, it is enough for me. More precisely the $x_1,x_2,\dots,x_n$ are probabilities so $x_i\in\left<0,1\right>$ and $\sum\limits_{i=1}^n x_i\in\left<0,1\right>$. – Waney Mar 17 '21 at 08:30

1 Answers1

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Question: "Are there something general which can be use for my problem?"

Answer: In general if $p_1,\dots,p_l \in k[x_1,\dots,x_n]$ are polynomials over a field $k$, you may consider the quotient ring $A:=k[x_1,\dots,x_n]/(p_1,\dots,p_l)$. Let $I$ denote the system of polynomial equations $p_1=0,\dots,p_l=0$. The following holds:

Lemma: The system $I$ has a finite set of solutions (in a finite extension of $k$) if and only if $\dim_k(A)<\infty$.

You'll find a "proof" of this Lemma at the following link:

When does a system of polynomial equations have infinitely many solutions?

user26857
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hm2020
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