Algebraic geometry has its origin in the study of systems of polynomial equations f (x, .
, x )=0, 1 1 n .
f (x, .
, x )=0.
r 1 n Here the f ? k X, .
, X ] are polynomials in n variables with coe?cients in a ?eld k.
i 1 n n Thesetofsolutionsisasubset V(f, .
, f) ofk .
Polynomialequationsareomnipresent 1 r inandoutsidemathematics, andhavebeenstudiedsinceantiquity.
Thefocusofalgebraic geometry is studying the geometric structure of their solution sets.
n If the polynomials f are linear, then V(f, .
, f ) is a subvector space of k.
Its i 1 r size is measured by its dimension and it can be described as the kernel of the linear n r map k ? k, x=(x, .
, x ) ? (f (x), .
, f (x)).
1 n 1 r For arbitrary polynomials, V(f, .
, f ) is in general not a subvector space.
To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +.
g f 1 1 r r is a linear combination of the f (with coe?cients g ? k T, .
, T ]), then we have i i 1 n V(f, .
, f)= V(g, f, .
Thus the set of solutions depends only on the ideal 1 r 1 r a? k T, .
, T ] generated by the f .
About the Author: Prof.
Ulrich G rtz, Institut f r Experimentelle Mathematik, Universit t Duisburg-Essen.
Essen.
Torsten Wedhorn, Institut f r Mathematik, Universit t Paderborn.
Ulrich G rtz, Institute of Experimental Mathematics, University Duisburg-Essen.
Torsten Wedhorn, Department of Mathematics, University of Paderborn.
| Following | If g g f g f 1 1 r r is a linear combination of the f (with coecients g k |
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| Author | Prof |
Vieweg+teubner Verlag Algebraic geometry: part i: schemes. with examples and exercises, paperback/ulrich gortz
