Zeros Of Quaternion Polynomials

This dissertation studies the theory of zeros of quaternion polynomials, includes the following main contents and conclusions.For one-sided quaternion polynomials, the relationship among point-multiplicity,extended-multiplicity, and spherical multiplicity has been studied, several methods for their calculations are obtained, and a theorem of the relationship between degree and multiplicity is proved; in the aspect of reconstruction, the conditions for zeros to determine polynomials are characterized, a concept about generalized zero is introduced, and a theorem which shows that there is a one-to-one correspondence between polynomials and tuples of generalized zeros is established; in the aspect of radical solvability, the definition of solvability in radicals is proposed and a conclusion which asserts that one-sided quaternion polynomials of degree one or degree two are solvable in radicals while the polynomials of higher degree are not solvable in radicals in general is built.For two-sided general quaternion polynomials, the following results are obtained:first, zeros can only be isolated, spherical, or circular; second, the number of connected components of zero set is at most eight; third, zero set can only be of one of the following cases: a union of a 2-sphere and a set of isolated zeros with cardinality ≤ 2, a union of a circle and a set of isolated zeros with cardinality ≤ 7, a non-empty discrete set with cardinality ≤ 8.For two-sided standard quaternion polynomials, formulas for computing zero set are provided and the above third point can be strengthened as: zero set can be only the conjugate class of some quaternion, a union of a circle and a set of isolated zeros with cardinality≤ 2, or a non-empty discrete set with cardinality ≤ 8. Finally, the essential number conjecture is considered, a negative answer to the conjecture is given, and the sharp bound for the essential number of zero sets of two-sided standard quaternion polynomials is pointed out.