\input zb-basic \input zb-ioport \iteman{io-port 00052407} \itemau{Walcher, Sebastian} \itemti{Algebras and differential equations.} \itemso{Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press, Inc. iv, 184 p. (1991).} \itemab Consider a differential equation $x'=p(x)$, with $p$ homogeneous quadratic, on a finite dimensional vector space $V$. Define a bilinear composition on $V$ by $xy={1\over 2} (p(x+y)-p(x)-p(y))$, Then $V$ with this multiplication, is a commutative, nonassociative algebra, and a linear, solution-preserving map between two differential equations of the type above is a homomorphism of the corresponding algebras. The first part (1-6) contains the fundamental theory and its application to some classes of examples. The basic definitions and properties are presented; then subalgebras and algebraic invariant subsets are studied with emphasis on semi-invariants. In the second part (7-10), the objects of investigation are subalgebras of the Lie algebra Pol $V$ of all polynomial vector fields on a vector space $V$. If a given homogeneous polynomial $p$ is contained in a large finite dimensional subalgebra of Pol $V$ or if its centralizer is large, it implies conditions on the algebra determined by $p$ that can be very strong. This connection between the inner structure of $p$ and its role in Pol $V$ is the main subject of interest. The transitive subalgebra and the differential equations related to it are discussed, and the centralizer of a given homogeneous element of Pol $V$ is studied. Classes of differential equations can thus be solved explicitly. \itemrv{M.Bertrand (Mont-Saint-Aignan)} \itemcc{} \itemut{derivations; automorphisms; Lie algebra of polynomial vector fields; homogeneous polynomial; centralizer; semi-invariants} \itemli{} \end