Files of this talk (zip)
Regular chains are one of the major tools for solving polynomial
systems. For systems of algebraic equations, they provide a convenient
way to describe complex solutions and a step toward isolation of real
roots or decomposition into irreducible components. Combined with other
techniques, they are used for these purposes by several computer algebra
For systems of partial differential equations, they provide a popular
way for determining a symbolic description of the solution set.
Moreover, thanks to Rosenfeld's Lemma, techniques from the algebraic
case apply to the differential one
In this talk, we first review the fundamental differential operations
that, in practice, rely directly on this reduction to the algebraic
case, namely pseudo-division, regularity test, regular GCDs and ranking
conversions in some cases. Then, we discuss how the recent improvements
of the algebraic operations (based on modular methods, fast polynomial
algorithms) can benefit to their differential counterparts.