This thesis explores the interplay between combinatorics, geometry, and algebraic systems through the lens of integer lattice structures and polynomial root counting. Beginning with the classical Frobenius problem, we investigate the space of linear Diophantine equations and demonstrate how solutions can be characterized geometrically using lattice polytopes. In the two-variable case, we derive the closed-form expression for the Frobenius number and build a foundation in the geometry of lattice polygons, later introducing Pick's formula to calculate the Euclidean area of any lattice polygon given its lattice points.

Extending to higher dimensions, we introduce Ehrhart theory as a framework for understanding the enumeration of lattice points in dilations of rational polytopes. We develop the Ehrhart series and its connections to generating functions, setting the stage for a deeper exploration into algebraic geometry. The focus then shifts to systems of polynomial equations and the Newton polytopes associated with them. We present Bernstein’s Theorem and its geometric proof via mixed volumes, revealing a profound connection between the combinatorics of Newton polytopes and the algebraic structure of polynomial systems in the algebraic torus.

Finally, we derive Bézout’s classical Theorem as a special case of the BKK bound and explore the emerging field of mixed Ehrhart theory, highlighting how it naturally bridges discrete geometry and algebraic complexity. Throughout, emphasis is placed on intuition, geometric interpretation, and the deep structural unification between counting, volume, and algebraic solutions.