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Algebraic Geometry
What is algebraic geometry? To start from something that you probably know, we can say that algebraic geometry is the combination of linear algebra and algebra:
• In linear algebra, we study systems of linear equations in several variables.
• In algebra, we study (among other things) polynomial equations in one variable.
Algebraic geometry combines these two fields of mathematics by studying systems of polynomial equations in several variables.
Given such a system of polynomial equations, what sort of questions can we ask? Note that we cannot expect in general to write down explicitly all the solutions: we know from algebra that even a single complex polynomial equation of degree d > 4 in one variable can in general not be solved exactly. So we are more interested in statements about the geometric structure of the set of solutions. For example, in the case of a complex polynomial equation of degree d, even if we cannot compute the solutions we know that there are exactly d of them (if we count them with the correct multiplicities). Let us now see what sort of “geometric structure” we can find in polynomial equations in several variables.
• In linear algebra, we study systems of linear equations in several variables.
• In algebra, we study (among other things) polynomial equations in one variable.
Algebraic geometry combines these two fields of mathematics by studying systems of polynomial equations in several variables.
Given such a system of polynomial equations, what sort of questions can we ask? Note that we cannot expect in general to write down explicitly all the solutions: we know from algebra that even a single complex polynomial equation of degree d > 4 in one variable can in general not be solved exactly. So we are more interested in statements about the geometric structure of the set of solutions. For example, in the case of a complex polynomial equation of degree d, even if we cannot compute the solutions we know that there are exactly d of them (if we count them with the correct multiplicities). Let us now see what sort of “geometric structure” we can find in polynomial equations in several variables.
133 pages, Unknown Binding
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