Existential decidability of a ring is the question as to whether an algorithm exists which determines whether a given system of polynomial equations and inequations has a solution. It is a classical result (``Hilbert's 10th problem'') that the ring of integers is not existentially decidable. Over the years there has been many results related to Hilbert 10th problem over different fields. For instance, the existential decidability of a Henselian valued field of mixed characteristic and finite ramification can be reduced to the positive existential decidability of its residue field, plus some additional structure.

An example of a mixed characteristic Henselian field is the fraction field of Witt Vectors. It is a construction analogous to the construction of the p-adic numbers from $\mathbb{F}_p$, and it takes a perfect field $F$ of characteristic $p$ and constructs a field with value group $\mathbb{Z}$ and residue field $F$. We will look at the existential decidability of the Henselian valued fields arising from finite extensions of the Witt vectors over a positive characteristic Henselian valued field. I will report on our progress so far, the problems that we have encountered, and the goals we are working toward.

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Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems.