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I'm interested in the topological torus, ie. the homeomorphism class of $S^1\times S^1$. Clearly, it can be realized as the real algebraic variety in $\mathbb{R}^4$ as the solution set to $x_1^2+x_2^2 = 1$, $x_3^2+x_4^2=1$. Now I wonder, is there a polynomial...