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Associate forms, joins, multiplicities and an intrinsic elimination theory. (English) Zbl 0762.13006
Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 71-108 (1990).
[For the entire collection see Zbl 0716.00007.]
The first time a mathematician hears about “multiplicity” $$m_ r\in\mathbb{Z}^ +$$, it refers to an $$m_ r$$-ple root of a polynomial $$f(x)$$ or binary form $$\varphi(x_ 0,x_ 1)$$: $$f(x)=a_ 0\prod_{r\in\mathbb{P}_ 1(\mathbb{C})}(x-r)^{m_ r}$$, $$\varphi(x_ 0,x_ 1)=\prod_{r\in\mathbb{P}_ 1(\mathbb{C})}\left|{x_ 0\atop r_ 0} {x_ 1\atop r_ 1}\right|^{m_ r}$$, $$m_ r\in\mathbb{Z}$$, $$m_ r\geq 0$$. It is natural to ask whether or not this exponent is also the natural intersection multiplicity of an irreducible component $$I$$ in the proper intersection $$V\cap W$$ $$(V,W$$ irreducible affine variety in $$\mathbb{P}_ n(\mathbb{C})=\mathbb{P}(E))$$. An affirmative answer is found in the book by B. L. van der Waerden: “Einführung in die algebraische Geometrie” (2nd edition 1973; Zbl 0264.14001) only for two irreducible plane curves. This idea of the exponent intersection multiplicity is developed in this paper in the general case by showing that the form $$F_{V\cdot W}=\prod F_ I^{m_ I}$$ (I proper irreducible component of $$V\cap W)$$ can be computed by restriction of the $$F_ J$$ (associated to the join $$J(V\times W)\subset\mathbb{P}(E\oplus E))$$ to the diagonal subspace $$\Delta\subset\mathbb{P}(E\oplus E)$$. The method extends naturally to $$h(\geq 2)$$ affine varieties $$V^{(j)}\subset\mathbb{P}_ n$$, $$j=1,\dots,h$$ provided $$c=\sum c_ j\leq n$$ $$(c_ j=\text{codim} V^{(j)}$$ in $$\mathbb{P}_ n)$$. The geometric interpretation of $$F_ V$$ in terms of the complex $${\mathfrak C}(V^ c)=\{\mathbb{P}_{c-1}\subset\mathbb{P}_ n|\mathbb{P}_{c-1}\cap V^ c\neq\emptyset\}$$ leads naturally to an equivalence of the exponent multiplicity with van der Waerden’s theory.
Since $$c=\text{codim} J$$ in $$\mathbb{P}(E\oplus\cdots\oplus E)$$ $$(h$$ copies of $$E)$$ a natural discussion arises also in the case $$c>n$$. Then the old elimination theory can be replaced by intrinsic constructions. Natural applications are made to Bézout’s theorem as well to possible future relations with the “length multiplicity”.

##### MSC:
 13H15 Multiplicity theory and related topics 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 13B25 Polynomials over commutative rings 14A05 Relevant commutative algebra