Information about Non Singular
In mathematics, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curve
exhibits at (0, 0), cannot simply be parametrized near the origin.
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
that make P the origin of coordinates. See Zariski tangent space for geometric and algebraic interpretation.
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" (Singular Points of Complex Hypersurfaces, pp. 12-13). Obviously the "geometric" meaning of "locally flat" over fields of finite characteristic, or ultrametric fields, is even more vexed.
- y2 = x2(x + 1)
exhibits at (0, 0), cannot simply be parametrized near the origin.
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
- F(x,y) = y2 − x2(x + 1),
so that the curve has equation
- F(x,y) = 0,
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
- F(x, y, z, ...) = 0
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
- Xi − Pi
that make P the origin of coordinates. See Zariski tangent space for geometric and algebraic interpretation.
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" (Singular Points of Complex Hypersurfaces, pp. 12-13). Obviously the "geometric" meaning of "locally flat" over fields of finite characteristic, or ultrametric fields, is even more vexed.
Singular points of smooth mappings
As the notion of singular points is a purely local property the above definition can extended to cover the wider class of smooth mappings, (functions from M to Rn where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the jets of the mapping. The k-th jet is the Taylor series of the mapping truncated at degree k and deleting the constant term.See also
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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algebraic variety is essentially a (finite or infinite) set of points where a polynomial (in one or more variables) attains, or a set of such polynomials all attain, a value of zero.
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In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.
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In mathematics, a plane curve is a curve in a Euclidian plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
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In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
applied to homogeneous coordinates [X:Y:Z
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- F(X,Y,Z) = 0
applied to homogeneous coordinates [X:Y:Z
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coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
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Differential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative.
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In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
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In the branch of mathematics called multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function.
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In mathematics, a hypersurface is some kind of submanifold.
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- For differential geometry usage, see glossary of differential geometry and topology.
- In algebraic geometry, a hypersurface in projective space of dimension n
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In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".
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coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally).
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In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must add the ring's multiplicative identity element (1) to itself to get the additive identity element (0); the ring is said to have
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In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives.
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In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain.
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singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor.
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