Problem 1
Use the precise definitions, state all hypotheses, apply bounded entire ⇒ constant, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Liouville’s Theorem and Fundamental Theorem of Algebra
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Liouville’s Theorem and Fundamental Theorem of Algebra.
Problem 4
The argument must address using bounded entire functions to prove polynomial root existence with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in evaluation of integrals, potential theory, analytic continuation, and spectral methods, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.