# Number of connected components of degree 2 affine algebraic varieties

Suppose an algebraic variety $$V$$ is given as the solutions to $$q$$ polynomial equations of degree $$\le k$$ with real coefficients $$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$ for $$x\in\mathbb R^m$$. It is a theorem due to Milnor that the sum of Betti numbers of $$V$$ is bounded by $$k(2k-1)^{m-1}$$. In particular this also bounds the number of connected components of $$V$$.

Question: Is this bound on the number of connected components of $$V$$ the best known bound? I am only interested in the case $$k=2$$. I suspect that for the varieties I am studying the number of connected components is bounded linearly in $$m$$.

Milnor, John W., On the Betti numbers of real varieties, Proc. Am. Math. Soc. 15, 275-280 (1964). ZBL0123.38302.

No. There are better bounds for particular cases (for instance for $$3$$ quadrics as given in the reference below), and in general it remains an open problem to find sharp bounds on the maximal number of components of a real affine/projective variety of given degree. The paper here of Degtyarev, Itenberg, and Kharlamov, answers some of the questions above.
Let $$B^0_r(N)$$, $$0\leq r\leq N − 1$$, denote the maximal number of connected components that a regular complete intersection of $$r + 1$$ real quadrics in $$\mathbb{P}^N_{\mathbb{R}}$$ can have. Then, it is straightforward to show that the number of components can grow exponentially in the dimension for a suitable number of quadrics
$$B^0_{N−1}(N) = 2^N$$ for all $$N > 1$$ (intersection of dimension zero).
However, if the number of equations is small, for instance three, then there is a quadratic upper bound in $$N$$ for $$B^0_{k}(N)$$.
Theorem. For all $$N > 4$$, one has $$\frac{1}{4}(N − 1)(N + 5) − 2 < B^0_2(N)\leq \frac{3}{2}k(k − 1) + 2$$, where $$k = [\frac{N}{2}] + 1$$.