A curve is the zero set of a polynomial
You have spent this whole ladder meeting curves one family at a time: the line y = mx + b, the circle x^2 + y^2 = r^2, the ellipse and its conic cousins. Step back and notice what they share. In every case you took a polynomial in x and y, set it equal to zero, and called the set of solution points 'the curve'. That single move — collect the points where one polynomial vanishes — is the whole definition of an algebraic curve. A line is where x - y = 0; a circle is where x^2 + y^2 - 1 = 0; the cubic where y^2 - x^3 - x = 0 is just as legitimate a curve, only wilder. We are no longer studying shapes one species at a time; we are studying the zero set of any polynomial f(x, y) = 0, all at once.
The single most important number attached to such a curve is its degree — the highest total power among the terms, where a term like x^2 y counts as degree 3 (you add the exponents). Degree 1 polynomials give lines; degree 2 gives the conics you met two rungs ago (circles, ellipses, parabolas, hyperbolas); degree 3 gives cubics like the one above, and so on. Degree is not cosmetic. It governs almost everything that follows: how many times the curve can cross a line, how many points pin it down, and — the headline of this guide — how two curves are forced to meet. When mathematicians stack many such zero sets together, in any number of variables, the resulting object earns the broader name algebraic variety; a plane curve is simply the friendliest variety of all.
Reading a curve off its equation
How do you picture a curve you have never seen, given only f(x, y) = 0? The honest first tools are the ones you already own from coordinate geometry. Find the intercepts by setting y = 0 (where it crosses the x-axis) and x = 0 (the y-axis). Test for symmetry: if replacing y by -y leaves the equation unchanged, the curve is mirror-symmetric across the x-axis. Then sample: pick a few x-values, solve for y, and plot. For the cubic y^2 = x^3 - x, setting y = 0 gives x^3 - x = 0, so x = -1, 0, 1 — three x-axis crossings — and because only y^2 appears, the whole picture is symmetric top-to-bottom. Already, with no new machinery, the shape begins to emerge from the symbols.
Here is a small surprise worth pausing on. That cubic y^2 = x^3 - x does not come out as one connected loop. For x between -1 and 0 the right side is positive so y is real, and for x greater than 1 it is positive again — but between 0 and 1 the right side goes negative, so there are no real points there at all. The curve splits into two separate pieces: a closed oval over [-1, 0] and an unbounded branch sailing off for x > 1. A single tidy polynomial has produced a disconnected curve. This is your first taste of how much richer algebraic curves are than the lines and circles that trained your intuition.
Singular points: where a curve crosses itself or stings
Most points on a nice curve are smooth: zoom in close enough and the curve looks like a straight line, with one clean tangent direction — exactly the regularity you demanded of curves a rung ago. But polynomials can do worse, and the bad points have a name: a singular point is a place where the curve fails to have a single well-defined tangent line. Two flavours dominate the gallery. A node is where the curve passes through one point twice and crosses itself, like the centre of a figure-eight: two tangent lines, not one. A cusp is a sharp spike where the curve comes to a point and reverses, with the two branches sharing a single squashed-together tangent.
two famous singularities at the origin (0, 0)
node (a self-crossing) : y^2 = x^2 (x + 1)
near 0 this behaves like y^2 = x^2, i.e. y = x and y = -x
-> two tangent lines through the origin, the curve crosses itself
cusp (a sharp spike) : y^2 = x^3
the two branches y = +sqrt(x^3) and y = -sqrt(x^3)
meet the x-axis tangentially -> one repeated tangent, a spike
Why fuss over these rare bad points? Because singular points are where the deepest theory either breaks or gets interesting, and because you can detect them mechanically. The test, which you will appreciate fully once you have multivariable calculus in hand, is this: a point on f(x, y) = 0 is singular exactly when both partial derivatives — the rate of change of f in the x-direction and in the y-direction — vanish there at the same time as f itself. When even one partial is nonzero, the curve has a clear tangent and the point is smooth. A curve with no singular points anywhere is called non-singular or smooth, and these well-behaved curves are the ones on which the cleanest theorems hold without asterisks.
Going projective so the count comes out right
Now for the trouble that forces the next idea. A line and a circle can meet in two points, one, or none — think of a line slicing through a circle, grazing it, or missing entirely. Two parallel lines meet nowhere. This raggedness is irritating: the number of intersection points keeps depending on luck and position. The cure is the move you first met in projective geometry. Pass to the projective plane by adjoining, to every direction, a single new point at infinity where all lines of that direction agree to meet. Parallel lines now cross — at their shared point at infinity — and missing intersections that escaped to infinity, or hid in complex numbers, are recaptured.
Three honest adjustments make the count exact, and it is worth naming them rather than waving them away. First, count over the complex numbers, not just the reals: a line truly does meet a circle in two points, but when it misses in the real picture, those two points have complex coordinates. Second, work in the projective plane so intersections at infinity are counted. Third, count each intersection with its multiplicity — a tangent line touches a curve at what should be regarded as two coincident points, so it counts as 2, not 1. With all three conventions in force, the chaos of 'sometimes two, sometimes none' collapses into a single rigid number.
Bezout's theorem: degree times degree
With those three conventions paid for, an astonishingly clean law appears. Bezout's theorem says: in the complex projective plane, a curve of degree m and a curve of degree n meet in exactly m times n points, counted with multiplicity. That is the entire statement. Two lines (degree 1 and 1) meet in 1 times 1 = 1 point — and now even parallel ones do, at infinity. A line and a conic (1 and 2) meet in 2 points; a line and a cubic in 3; two conics (2 and 2) in 4; two cubics in 9. The ragged real-world count — sometimes two, sometimes none — was never the truth. The truth is the product of the degrees, always, with no exceptions once you honour the three conventions.
- Read off the two degrees m and n from the highest total power in each polynomial.
- Move to the complex projective plane so intersections that are complex or at infinity are not lost.
- Count each intersection with multiplicity — a tangency counts as 2, a higher-order touch as more.
- The total is exactly m times n; if you see fewer real points, the rest are complex, at infinity, or hidden inside a tangency.
Bezout also explains a fact you have used without proof. A conic has degree 2, and its general equation has six coefficients — but scaling them all by a constant gives the same curve, so really there are five free ratios to pin down. Each point the conic must pass through imposes one linear condition on those ratios. Five points in general position therefore determine exactly one conic, which is the meaning of five points determine a conic. The degree governed the head-count of conditions, just as it governed the intersection count — the same number doing double duty, which is exactly the kind of unity that makes algebraic geometry feel less like a bag of tricks and more like a single deep machine.
What you carry forward
Be honest about the scope of all this. We have stated Bezout's theorem and made it believable, but a full proof — pinning down multiplicity rigorously and proving the count never wavers — needs machinery (resultants, or the algebra of polynomial ideals) that lives beyond this introductory rung. That is no embarrassment; it is the normal shape of real mathematics, where a clean statement often rests on deep foundations. What you genuinely own now is the idea: degree is destiny, singular points are the exceptions to watch, and the projective–complex–multiplicity trio is the price of a count that finally behaves.
This algebraic view of geometry is not a museum piece. The cubic curves with no singular points, viewed over suitable number systems, are elliptic curves — the same objects at the heart of modern cryptography and of Wiles's proof of Fermat's Last Theorem. The varieties hinted at here are the daily language of contemporary algebraic geometry, one of the most active fields in mathematics. And the very next guides in this rung pivot to a different modern frontier — convexity and polytopes, then geometry done inside a computer, where curves and points become data and the theorems become algorithms.