A negative answer to a question about leading terms ideals of polynomial ideals

We construct an example of a finitely generated ideal I of , where is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if is a domain with Krull dimension ≤1, then for any finitely generated ideal I of , the leading terms ideal of I is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain and any , the ideal of generated by the leading terms of 〈1+aX,b〉 is finitely generated.