On the Problem of Separating Variables in Multivariate Polynomial Ideals

For a given ideal <Formula format="inline"><TexMath><?TeX I⊆ K[x₁,,xₙ,y₁,,yₘ]?></TexMath><AltText>Math 1</AltText><File name="issac24-10-inline1" type="svg"/></Formula> in a polynomial ring with n + m variables, we want to find all elements that can be written as f − g for some <Formula format="inline"><TexMath><?TeX f∈ K[x₁,,xₙ]?></TexMath><AltText>Math 2</AltText><File name="issac24-10-inline2" type="svg"/></Formula> and some <Formula format="inline"><TexMath><?TeX g∈ K[y₁,,yₘ]?></TexMath><AltText>Math 3</AltText><File name="issac24-10-inline3" type="svg"/></Formula>, i.e., all elements of I that contain no term involving at the same time one of the x1, …, xn and one of the y1, …, ym. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.