We prove that the positive-existential theory of addition and divisibility in a ring of polynomials in two variables A[t₁,t₂] over an integral domain A is undecidable and that the universal-existential theory of A[t₁] is undecidable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-2, author = {Thanases Pheidas}, title = {Diophantine undecidability for addition and divisibility in polynomial rings}, journal = {Fundamenta Mathematicae}, volume = {184}, year = {2004}, pages = {205-220}, zbl = {1058.03019}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-2} }
Thanases Pheidas. Diophantine undecidability for addition and divisibility in polynomial rings. Fundamenta Mathematicae, Tome 184 (2004) pp. 205-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-2/