Hyperbolic Differential Polynomials

I. Generalities.- I.1. Emission Cones.- I.2. The Topological Algebra D?(?).- I.3. The Set U(?) of Polynomial Distributions with Inverse in D?(?).- I.4. Bounded Subsets of U(?) with Bounded Inverse.- I.5. First Consequences of U being Invertible in D?(?)with Bounded Inverse.- 1.6. Remarks.- II. The Semi-algebraic Case. Criterion forUto be Invertible with Bounded Inverse.- II.1. Semi-algebraic Subsets of ?n.- II.2. Polynomial Mappings of ?n into ?m. Theorem of Seidenberg.- II.3. Asymptotic Behavior of Semi-algebraic Subsets of ?2.- II.4. If U is Invertible with Bounded Inverse, then the Union of the V(a) can be Localized.- II.5. Hyperbolicity of A or of $$bigcuplimits_{a in A} {Vleft( a right)}$$ the V(a). Criterion for U to be Invertible with Bounded Inverse.- II.6. Differential Polynomials that are a Polynomial Function of a Parameter (? ? ?p).- III. A Sufficient Condition thatUis Invertible with Bounded Inverse. The Cauchy Problem in Hsloc.- III.1. Upper Bounds for |a(?)|?1.- III.2. Laplace Transforms and Supports of Distributions.- III.3. A Sufficient Condition that U is Invertible with Bounded Inverse.- III.4. The Cauchy Problem with Data in Hsloc.- IV. Hyperbolic Hypersurfaces and Polynomials.- IV.0. Preliminary Notations and Definitions.- IV.1. First Properties of 0?-Hyperbolic V(a).- IV.2. First Properties of 0?-Hyperbolic Cones V(am).- IV.3. 0?-Hyperbolicity and ?-Hyperbolicity.- IV.4. Polars with respect to ? ? 0? of 0?-Hyperbolic V(a).- IV.5. Successive Multiplicities of a Series in ?[[X]] with respect to a Polynomial with Roots in ?[[X]]..- IV.6. Relations between V(am?k) and V(am) that follow from Hyperbolicity of V(a).- IV.7. Relations between V(am?k) and the Polars of V(am) Implied by Hyperbolicity of V(a).- IV.8. Functions Rm?k, ? on V(am(k), ?) ? ?n. A Sufficient Condition that V(a) is Hyperbolic.- IV.9. Local Properties of the Functions Rm?k, ?.- IV.10. Real Ordered Sheets of Hyperbolic Cones.- IV.11. Locally Constant Multiplicity on V(am)? ?n. A