# Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. Is there a monomial order on $R$ with a Groebner basis of degree $\leq e$ for $I$?

• What exactly do you mean by the degree of a Gröbner basis? – Neil Strickland May 8 '17 at 10:37