T1 space

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. Template:Distinguish Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees. This statement must be qualified in several important ways, by considering points at infinity, allowing complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), assigning an appropriate multiplicity to each intersection point, and excluding a degenerate case when X and Y have a common component. A simpler special case is when one does not care about multiplicities and X and Y are two algebraic curves in the Euclidean plane whose implicit equations are polynomials of degrees m and n without any non-constant common factor; then the number of intersection points does not exceed mn.

Bézout's theorem refers also to the generalization to higher dimensions: Let there be n homogeneous polynomials in Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. variables, of degrees $d_{1},\ldots ,d_{n}$, that define n hypersurfaces in the projective space of dimension n. If the number of intersection points of the hypersurfaces is finite over an algebraic closure of the ground field, then this number is $d_{1}\cdots d_{n},$ if the points are counted with their multiplicity. As in the case of two variables, in the case of affine hypersurfaces, and when not counting multiplicities nor non-real points, this theorem provides only an upper bound of the number of points, which is often reached. This is often referred to as Bézout's bound.

Bézout's theorem is fundamental in computer algebra and effective algebraic geometry, by showing that most problems have a computational complexity that is at least exponential in the number of variables. It follows that in these areas, the best complexity that may be hoped for will occur in algorithms have a complexity which is polynomial in Bézout's bound.

Rigorous statement

Suppose that X and Y are two plane projective curves defined over a field F that do not have a common component (this condition means that X and Y are defined by polynomials, whose polynomial greatest common divisor is a constant; in particular, it holds for a pair of "generic" curves). Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

The generalization in higher dimension may be stated as:

Let n projective hypersurfaces be given in a projective space of dimension n over an algebraic closed field, which are defined by n homogeneous polynomials in n + 1 variables, of degrees $d_{1},\ldots ,d_{n}.$ Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product $d_{1}\cdots d_{n}.$ If the hypersurfaces are irreducible and in relative general position, then there are $d_{1},\ldots ,d_{n}$ intersection points, all with multiplicity 1.

There are various proofs of this theorem. In particular, it may be deduced by applying iteratively the following generalization: if V is a projective algebraic set of dimension $\delta$ and degree $d_{1}$, and H is a hypersurface (defined by a polynomial) of degree $d_{2}$, that does not contain any irreducible component of V, then the intersection of V and H has dimension $\delta -1$ and degree $d_{1}d_{2}.$ For a (sketched) proof using the Hilbert series see Hilbert series and Hilbert polynomial#Degree of a projective variety and Bézout's theorem.

History

Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.

Intersection multiplicity

Template:See The most delicate part of Bézout's theorem and its generalization to the case of k algebraic hypersurfaces in k-dimensional projective space is the procedure of assigning the proper intersection multiplicities. If P is a common point of two plane algebraic curves X and Y that is a non-singular point of both of them and, moreover, the tangent lines to X and Y at P are distinct then the intersection multiplicity is one. This corresponds to the case of "transversal intersection". If the curves X and Y have a common tangent at P then the multiplicity is at least two. See intersection number for the definition in general.

Examples

• Two distinct non-parallel lines always meet in exactly one point. Two parallel lines intersect at a unique point that lies at infinity. To see how this works algebraically, in projective space, the lines x+2y=3 and x+2y=5 are represented by the homogeneous equations x+2y-3z=0 and x+2y-5z=0. Solving, we get x= -2y and z=0, corresponding to the point (-2:1:0) in homogeneous coordinates. As the z-coordinate is 0, this point lies on the line at infinity.

• The special case where one of the curves is a line can be derived from the fundamental theorem of algebra. In this case the theorem states that an algebraic curve of degree n intersects a given line in n points, counting the multiplicities. For example, the parabola defined by y - x² = 0 has degree 2; the line y − ax = 0 has degree 1, and they meet in exactly two points when a ≠ 0 and touch at the origin (intersect with multiplicity two) when a = 0.

• Two conic sections generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:

• • Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four. The discrepancy comes from the fact that every circle passes through the same two complex points on the line at infinity. Writing the circle

$(x-a)^{2}+(y-b)^{2}=r^{2}$

in homogeneous coordinates, we get

$(x-az)^{2}+(y-bz)^{2}-r^{2}z^{2}=0,$

from which it is clear that the two points (1:i:0) and (1:-i:0) lie on every circle. When two circles don't meet at all in the real plane (for example because they are concentric) they meet at exactly these two points on the line at infinity with an intersection multiplicity of two.

• • Any conic should meet the line at infinity at two points according to the theorem. A hyperbola meets it at two real points corresponding to the two directions of the asymptotes. An ellipse meets it at two complex points which are conjugate to one another---in the case of a circle, the points (1:i:0) and (1:-i:0). A parabola meets it at only one point, but it is a point of tangency and therefore counts twice.
  • The following pictures show examples in which the circle x²+y²-1=0 meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than 1:

$x^{2}+4y^{2}-1=0:\ {\hbox{two intersections of multiplicity 2}}$

$5x^{2}+6xy+5y^{2}+6y-5=0:\ {\hbox{an intersection of multiplicity 3}}$

$4x^{2}+y^{2}+6x+2=0:\ {\hbox{an intersection of multiplicity 4}}$

Sketch of proof

Write the equations for X and Y in homogeneous coordinates as

$a_{0}z^{m}+a_{1}z^{m-1}+\dots +a_{m-1}z+a_{m}=0$

$b_{0}z^{n}+b_{1}z^{n-1}+\dots +b_{n-1}z+b_{n}=0$

where a_i and b_i are homogeneous polynomials of degree i in x and y. The points of intersection of X and Y correspond to the solutions of the system of equations. Form the Sylvester matrix; in the case m=4, n=3 this is

$S={\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0&0\\0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0\\0&0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\b_{0}&b_{1}&b_{2}&b_{3}&0&0&0\\0&b_{0}&b_{1}&b_{2}&b_{3}&0&0\\0&0&b_{0}&b_{1}&b_{2}&b_{3}&0\\0&0&0&b_{0}&b_{1}&b_{2}&b_{3}\\\end{pmatrix}}.$

The determinant |S| of S, which is also called the resultant of the two polynomials, is 0 exactly when the two equations have a common solution in z. The terms of |S|, for example (a₀)ⁿ(b_n)^m, all have degree mn, so |S| is a homogeneous polynomial of degree mn in x and y (recall that a_i and b_i are themselves polynomials). By the fundamental theorem of algebra, this can be factored into mn linear factors so there are mn solutions to the system of equations. The linear factors correspond to the lines that join the origin to the points of intersection of the curves.

See also

• AF+BG theorem

References

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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Alternative translation of earlier (2nd) edition of Newton's Principia.

• (generalization of theorem) http://mathoverflow.net/questions/42127/generalization-of-bezouts-theorem

External links

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• Bezout's Theorem at MathPages