# Seidenberg A.'s Basic theorems in partial differential algebra PDF

By Seidenberg A.

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Although they can be deﬁned for arbitrary base pointed manifolds (M, x0 ) we present the theory only in the case χ(M ) = 0 when the base point is irrelevant. While the complex Ray–Singer torsion and dynamical torsion are new concepts the Milnor–Turaev torsion is not, however our presentation is somehow different from the traditional one. In Section 4 we discuss the algebraic variety of cochain complexes of ﬁnite-dimensional vector spaces and introduce the Milnor torsion as a rational function on this variety.

53 6 Milnor–Turaev and dynamical torsion . . . . . . . . . . . . . . . . . 57 7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1. Introduction For a ﬁnitely presented group Γ denote by Rep(Γ; V ) the algebraic set of all complex representations of Γ on the complex vector space V .

62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1. Introduction For a ﬁnitely presented group Γ denote by Rep(Γ; V ) the algebraic set of all complex representations of Γ on the complex vector space V . For a closed base pointed manifold (M, x0 ) with Γ = π1 (M, x0 ) denote by RepM (Γ; V ) the algebraic closure of RepM 0 (Γ; V ), the Zariski open set of representations ρ ∈ Rep(Γ; V ) so that H ∗ (M ; ρ) = 0.

### Basic theorems in partial differential algebra by Seidenberg A.

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