Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov

By Ivan Soprunov

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55. (a) The nodal cubic has equation f (x, y) = x2 − y 2 + x3 = 0. We see that f0 = f1 = 0, f2 = x2 − y 2 , and f3 = x3 . Therefore (0, 0) is a double point. (b) The cuspidal cubic has equation f (x, y) = y 2 − x3 = 0, hence f0 = f1 = 0, f2 = y 2 , and f3 = −x3 . Again (0, 0) is a double point. (c) Consider the affine curve with equation y 2 = x3 − x. This time f (x, y) = x + y 2 − x3 , hence, f0 = 0 and f1 = x. Therefore, (0, 0) is a smooth point of the curve. 8). 8. The smooth cubic y 2 = x3 − x.

Then y n − f (x) is absolutely irreducible. For example, y n − x is absolutely irreducible. Note a striking distinction between univariate and multivariate case. When F is algebraically closed, the only irreducible univariate polynomials are linear, whereas there are absolutely irreducible multivariate polynomials of arbitrarily large degree. 2. 2. 1. Affine Curves. Let F be a field. We call the Cartesian power Fn = {(x1 , . . , xn ) | xi ∈ F} the affine space over F and denote by AnF or, simply, by An .

You should show that the two definitions are equivalent. Although we will mostly be dealing with bivariate polynomials we will give the general definition of the multivariate polynomial ring. 16. Let R be a commutative ring with 1. Define a n-variate polynomial f (x1 , . . , xn ) over R as a finite linear combinations of monomials xi11 · · · xinn with coefficients in R: � f (x1 , . . ,in are zero. The set of all polynomials f (x1 , . . , xn ) forms the ring of polynomials R[x1 , . . , xn ] under usual operations of addition and multiplication.

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