Approximate Commutative Algebra From Oil Fields to Hilbert Schemes Numerical Decomposition of the Rank Deficiency Set of a Matrix of Multivariate Polynomials Towards Geometric Completion of Differential Systems by Points Geometric Involutive Bases and Applications to Approximate Commutative Algebra Regularization and Matrix Computation in Numerical Polynomial Algebra Ideal Interpolation: Translations to and from Algebraic Geometry An Introduction to Regression and Errors in Variables from an Algebraic Viewpoint ApCoA = Embedding Commutative Algebra into Analysis Exact Certification in Global Polynomial Optimization Via Rationalizing Sums Of Squares. Approximate Commutative Algebra is an emerging field of research which endeavours to bridge the gap between traditional exact Computational Commutative Algebra and approximate numerical computation. The last 50 years have seen enormous progress in the realm of exact Computational Commutative Algebra and given the importance of polynomials in scientific modelling it is very natural to want to extend these ideas to handle approximate empirical data deriving from physical measurements of phenomena in the real world. In this volume nine contributions from established researchers describe various approaches to tackling a variety of problems arising in Approximate Commutative Algebra.