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In this paper, we study ideals spanned by polynomials or overconvergentseries in a Tate algebra. With state-of-the-art algorithms for computing TateGr{\"o}bner bases, even if the input is polynomials, the size of the outputgrows with the required precision, both in terms of the size of thecoefficients and the size of the support of the series. We prove that idealswhich are spanned by polynomials admit a Tate Gr{\"o}bner basis made ofpolynomials, and we propose an algorithm, leveraging Mora's weak normal formalgorithm, for computing it. As a result, the size of the output of thisalgorithm grows linearly with the precision. Following the same ideas, wepropose an algorithm which computes an overconvergent basis for an idealspanned by overconvergent series. Finally, we prove the existence of auniversal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras,compatible with all convergence radii.

On Polynomial Ideals And Overconvergence In Tate Algebras