A Gröbner basis (Buchberger 1965) is a special generating set for a polynomial ideal that makes division uniquely defined — analogous to how row reduction gives a unique echelon form for linear systems. The algorithm works by computing S-polynomials (differences canceling leading terms) and reducing them to zero, iterating until closure. The choice of monomial order (lex, grlex, grevlex) dramatically affects both the basis and computation time. Elimination theory: with lex order, a Gröbner basis automatically eliminates variables one-by-one, turning a system of polynomial equations into a single univariate polynomial solvable by radicals. This visualization shows the zero sets (algebraic curves) of two polynomials and their intersection points — exactly what Gröbner bases compute.