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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 6

Let \(p(z)=a_nz^n+\cdots +a_0\) be a non-constant complex polynomial. Prove \(p:\mathbb {C}\to \mathbb {C}\) is proper, has only a finite number of singular values, that the inverse images of regular values have constant cardinality. Deduce the Fundamental Theorem of Algebra: \(p\) has a root.

Raw LaTeX

Let \(p(z)=a_nz^n+\cdots+a_0\) be a non-constant complex polynomial. Prove
\(p:\mathbb{C}\to\mathbb{C}\) is proper, has only a finite number of singular values,
that the inverse images of regular values have constant cardinality. Deduce the
Fundamental Theorem of Algebra: \(p\) has a root.

Topic Tags

Fundamental Groups Lie Algebras and Exponential Maps Regular Values and Level Sets