exhibits at (0, 0), cannot simply be parametrized near the origin. A plot of this curve is below with the singular point at the origin. Singular points commonly occur where a graph crosses over itself:
The reason for that algebraically is that both sides of the equation show powers higher than 1 of the variables x and y. In terms of differential calculus, if
so that the curve has equation
then the partial derivatives of F with respect to both x and y vanish at (0,0). This means that if we try to use the implicit function theorem to express y as a function of x near y = 0, we shall fail; and indeed no linear combination of x and y is a function of another essentially different one, so that this is a geometric condition not tied to any choice of coordinate axes.
In general for a hypersurface
the singular points are those at which all the partial derivatives simultaneously vanish. A general algebraic variety V being defined by several polynomials, or in algebraic terms an ideal of polynomials, the condition on a point P to be a singular point of V is that none of those polynomials have a non-zero linear (degree 1) term, when written in terms of variables
Points of V that are not singular are non-singular. Apart from some technical questions that can be caused by non-zero characteristic, it is always true that most points are non-singular.
It is important to note that the geometric criterion for a point of a variety to be singular (mentioned earlier), that it is a point where the variety is not "locally flat", can be very hard to recognize for varieties over a general field. The work of Milnor and others shows that, over the complex numbers, the statement is precisely true in every reasonable interpretation. But, as Milnor points out, over the real numbers "The equation ... can actually be solved for as a real analytic function of " (so that the variety it defines is the graph of a real analytic function, and therefore a real analytic manifold) "but this equation also defines a variety having a singular point at the origin". Obviously the "geometric" meaning of "locally flat" over fields of finite characteristic, or ultrametric fields, is even more vexed.
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