I wish to first present a general approach to preserving and exploiting symmetry in algebraic computations that boil down to linear algebra in finite dimensional vector spaces of polynomials. This will be illustrated on the easily understood problem of multivariate interpolation, though the underlying principle applies to global optimisation, computing cubatures, as well as computations in physics and chemistry.... The key observation is that the matrices involved become block diagonal, with repeated blocks, in a symmetry adapted bases.

Multivariate Lagrange and Hermite interpolation are examples of ideal interpolation. More generally an ideal interpolation problem is defined by a set of linear forms, on the polynomial ring, whose kernels intersect into an ideal.

The article addresses multivariate interpolation in the presence of symmetry. Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases. This allows to construct bases of invariant interpolation spaces in blocks, capturing the inherent redundancy in the computations. We show that the so constructed symmetry adapted interpolation bases alleviate the computational cost of any interpolation problem and automatically preserve any equivariance of this interpolation problem might have.

Scalings form a class of group actions that have both theoretical and practical importance. A scaling is accurately described by an integer matrix. Tools from linear algebra are exploited to compute a minimal generating set of rational invariants, trivial rewriting and rational sections for such a group action. The primary tools used are Hermite normal forms and their unimodular multipliers. With the same line of ideas, a complete solution to the scaling symmetry reduction of a polynomial system is also presented.

We review the algebraic foundations we developed to work with differential invariants of finite dimensional group actions. Those support the algorithms we introduced to operate symmetry reduction with a view towards differential elimination.

We introduce new ideas to improve the efficiency and rationality of a triangulation decomposition algorithm. On the one hand we identify and isolate the polynomial remainder sequences in the triangulation-decomposition algorithm. Subresultant polynomial remainder sequences are then used to compute them and their specialization properties are applied for the splittings. The gain is two fold: control of expression swell and reduction of the number of splittings. On the other hand, we remove the role that initials had in previous triangulation-decomposition algorithms. They are not needed in theoretical results and it was expected that they need not appear in the input and output of the algorithms. This is the case of the algorithm presented. New algorithms are presented to compute a subsequent characteristic decomposition from the output of the triangulation decomposition algorithm where the initials need not appear.

This paper presents a new algorithm to compute the power series solutions of a significant class of nonlinear systems of partial differential equations. The algorithm is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equations to get coefficients of the power series, one at a time. The algorithm presented here relies on using the linearisation of the system and the associated recurrences. At each step the order up to which the power series solution is known is doubled. The algorithm can be seen as belonging to the family of Newton iteration methods.

We consider in this paper an implicit non-linear ordinary differential equation P(Z, y, y’) = O. There are several types of solutions. The singular solutions are characterized by the fact they make & vanish. No such characterization of the so called general solution can be found in classical treatises. We propose here an algorithmic method to compute a similar characterisation of the general solution. We apply the result to give some insight on the local behaviour of the solutions in the neighbourhood of a singular solution.