An effective construction of a small number of equations defining an algebraic variety

Consider a system of polynomial equations in $n$ variables of degrees at most $d$ with the set of all common zeros $V$. We suggest subexponential time algorithms (in the general case and in the case of zero characteristic) for constructing $n+1$ equations of degrees at most $d$ defining the algebraic variety $V$.
Further, we construct $n$ equations defining $V$. We give an explicit upper bound on the degrees of these $n$ equations. It is double exponential in $n$. The running time of the algorithm for constructing them is also double exponential in $n$.