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Added higher-order tag. Minor edits.
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edit approved Jul 10, 2020 at 21:38
Rodrigo de Azevedo
edit approved Jul 10, 2020 at 21:38
Rodrigo de Azevedo
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Suppose we have two polynomials p,q in $\mathbb{R}^3$$p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional curve. But how can I make this statement rigourous? Is there any theorem which gives you the hausdorffHausdorff dimension of such a zero set? Indeed, we will have to assume that p$p$ and q$q$ have no common factors.

Suppose we have two polynomials p,q in $\mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional curve. But how can I make this statement rigourous? Is there any theorem which gives you the hausdorff dimension of such a zero set? Indeed, we will have to assume that p and q have no common factors.

Suppose we have two polynomials $p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional curve. But how can I make this statement rigourous? Is there any theorem which gives you the Hausdorff dimension of such a zero set? Indeed, we will have to assume that $p$ and $q$ have no common factors.

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mathuser
mathuser
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Finding the dimension of the intersection of two real algebraic varieties

Suppose we have two polynomials p,q in $\mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional curve. But how can I make this statement rigourous? Is there any theorem which gives you the hausdorff dimension of such a zero set? Indeed, we will have to assume that p and q have no common factors.