Taken from my report for a Computer Algebra course.
We know there are plenty of methods to solve a system of linear equations (to name a few: Gauss elimination, QR or LU factorization). In fact, it is straightforward to check whether a linear system has any solutions, and if it does, how many of them there are. But what if the system is made of non-linear equations? The invention of Groebner bases and the field of computational algebra came up to answer these questions.
In this text we will recap the theory behind single-variable polynomials and extend it to multiple-variable ones, ultimately getting to the definition of Groebner bases.
In some cases, the transition from one to multiple variables is smooth and pretty much an extension of the simple case (for example for the Greatest Common Divisor algorithm). In other cases, however, there are conceptual jumps to be made. To give an example, single variable polynomials have always a finite number of roots, while this does not hold for multivariable polynomials. Intuitively, the reason is that a polynomial in one variable describes a curve in the plane, which can only intersect the x-axis a discrete and finite number of times. On the other hand, a multivariate polynomial describes a surface in space, which will always intersect the 0-plane in a continuum of points.