Given $n$ quadratic polynomials in $n-1$ variables over the complex field, what is the maximum number of common zeros? Can we have $2^{n-1}-1$ common zeros? I assume that a linear combination of the polynomials is always different from zero and the number of zeros is finite.

With $4$ polynomials, the maximum is not smaller than $6$. Using a projective space $(x_1,x_2,x_3,x_4)$, an example with $6$ roots is given by the polynomials.

$P_1=x_1 x_2$,

$P_2=x_1 x_3$,

$P_3=L_1 x_2+L_2 x_3$,

$P_4=(\text{a general quadratic polynomial})$,

$L_k$ being general linear polynomials.
Indeed, if $x_1=0$, then the first two polynomials are equal to zero and the remaining two polynomials in $x_2, x_3, x_4$ give four roots, which are distinct for a general choice of $L_k$ and $P_4$. If $x_2=x_3=0$, then the first three polynomials are equal to zero and the fourth polynomial in $x_1,x_4$ gives 2 additional roots.

This construction has a natural generalisation to $n$ polynomials, giving $2^{n-2}+2^{n-3}$ roots, which is about $3/4$ of the desired bound $2^{n-1}-1$.

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