## Honors Theses

4-27-2004

Thesis

Bachelor of Arts

#### Department

Mathematics

Dr. William T. Ross

#### Abstract

In this project, we address the question: When can a polynomial p(x, y) of two variables be factored as p(x, y) = f(x)g(y), where f and g are polynomials of one variable. We answer this question, using linear algebra, and create a Mathematica program which carries out this factorization. For example,

3+3x-5x^3+y+xy-5/3x^3y+y^2+xy^2-5/3x^3y^2 = (1+x-5/3x^3)(3+y+y^2)

We then generalize this concept and ask: When can p(x,y) can be written as

p(x,y) = f1(x)g2(y)+f2(x)g2(y)+...+fr(x)gr(y)

where fj,gj are polynomials. This can certainly be done (for large enough r). Which is the minimum such r? Again, we have a Mathematica program which carries out this computation. For example,

1+2x+x^2+2x^3+2y+2x^2y+7xy^2+7x^3y^2=(1+x^2)(1+2y)+(x+x^3)(2+7y^2)

We generalize this further to larger number of variables (with an appropriate Mathetmatica program to carryout this computation). We then apply this and consider the domains of convergence of certain types of real analytic functions and try to relate the domain of convergence with the rank of the polynomial.

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