Corollary 2

Let K be a commutative field, $n$ be a natural number, $(P_i)_{i\in I}$ be a family of elements of $K[X_1, . . . ,X_n]$, and Ω be an algebraically closed extension of K. The following conditions are equivalent:

(i) The polynomials $P_i$ do not have a common zero in $\Omega^n$.

(ii) There exists a family $(Q_i)_{i\in I}$, with finite support, of elements of $K[X_1, . . . ,X_n]$ such that $\sum_{i\in I}P_iQ_i= 1$.

Denote by A the ring $K[X_1, . . . ,X_n]$ and by $\mathfrak{a}$ the ideal generated by the polynomial $P_i$. Condition (i) means that there are no K-algebra homomorphisms from $A/\mathfrak{a}$ to Ω. If it is satisfied, then by Corollary 1, the ring $A/\mathfrak{a}$ has no maximal ideals and is therefore zero, and 1 belongs to $\mathfrak{a}$. This proves that (i) implies (ii). The implication (ii) $\Rightarrow$ (i) is clear.