Abstract
Let ( A , m A , k ) be a local noetherian ring and I an m A -primary ideal. The asymptotic Samuel function (with respect to I ) v I ¯ : A ⟶ R ∪ { + ∞ } is defined by v I ¯ ( x ) = l i m k → + ∞ o r d I ( x k ) k , ∀ x ∈ A . Similarly, one defines, for another ideal J , v I ¯ ( J ) as the minimum of v I ¯ ( x ) as x varies in J . Of special interest is the rational number v I ¯ ( m A ) . We study the behavior of the asymptotic Samuel function (with respect to I ) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.
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