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| {{About|Bézout's theorem in algebraic geometry|Bézout's theorem in arithmetic|Bézout's identity}}
| | Prostatic congestion and inflammation can stimulate the prostate urethra inducing frequent spermatorrhea. Increase Testosterone Level - For a long time it was thought that certain unwanted symptoms were just part of the aging process and that there was nothing that could be done to treat them. Treatments or medications used for prostrate cancer reduce the manufacture of testosterone in the male testes and adrenal glands. There are many reasons that business ethics matter. Most of these herbs have been in use for centuries, used by Africans, Asians, Americans etc. <br><br>Here are some remedies for impotence you can try tonight. Impotence (or erectile dysfunction) is defined as the inability to achieve or maintain an erection sufficient for sexual intercourse, and includes the inability to get an erection as a result of sexual stimulation or to lose your erection prior to ejaculation. In a published study reported in the March issue of Urology, researchers surveyed over 1,000 men who underwent radical prostatectomy (RP) between 1962 and 1997 (85. A few of the patterns truly flow straight down to the brim of the cap, although some have add-ons decorating the top like organizations or metallic studs, and sometimes several rhinestones. To stay healthy, it is critical to get peaceful rest for not less than 8 hrs on a daily basis. <br><br>Did you know that the way you breathe also has a major impact upon your impotency problems. For a male member to be total and erect, it will get lots of blood stream which makes growing the erection tissue inside it. If we talk about an effective and reliable treatment for erectile dysfunction, then the use of herbs would be the most appropriate. Though the previous studies conducted on the potential aspects of antioxidants were not that conclusive, but that other data indicates nutritional therapies may have significant potential. British researchers said pelvic floor exercises or Kegels, named after the South Californian gynecologist Dr. <br><br>Try to be supportive but don't let the issue go unresolved as it has the potential to shake the foundation of your relationship if not addressed. His partner also supports this action, which also helps in getting his sex life back on track. Spinal cord injury, old age, low level of testosterone hormone, depression and high stress are some common causes leading way to this reproductive disorder. You should also be eliminating your body and your bloodstream. For guys, these lotions work just for about 5 percent of men and are difficult to use. <br><br>Mutations in the HFE, SLC40A1, HAMP, TFR2, and HFE2 genes cause this genetic disorder. Some of these habits include excessive drinking and smoking. Simply applying a male organ health cream (health professionals recommend Man 1 Man Oil) directly to the male organ after a shower will deliver important nutrients directly to the male organ where it is needed most. This is why Erectile Dysfunction Drugs were created. Follow the instructions on container for the required dosage.<br><br>In case you have virtually any issues with regards to wherever along with how to use over the counter erectile dysfunction pills [[http://www.eiaculazione-precoce.info/sitemap/ navigate to this web-site]], it is possible to e mail us on our web page. |
| {{distinguish|Little Bézout's theorem}}
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| '''Bézout's theorem''' is a statement in [[algebraic geometry]] concerning the number of common points, or intersection points, of two plane [[algebraic curve]]s. The theorem claims that the number of common points of two such curves ''X'' and ''Y'' is equal to the product of their [[degree of a polynomial|degree]]s. 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 [[intersection number|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
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| ''Y'' are two [[algebraic curve]]s in the [[Euclidean plane]] whose [[implicit equation]]s are [[polynomial]]s of degrees ''m'' and ''n'' without any non-constant common factor; then the number of intersection points does not exceed ''mn''.
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| '''Bézout's theorem''' refers also to the generalization to higher dimensions: Let there be ''n'' [[homogeneous polynomial]]s in {{math|''n''+1}} variables, of degrees <math>d_1, \ldots, d_n</math>, that define ''n'' [[hypersurface]]s 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 <math>d_1 \cdots d_n,</math> if the points are counted with their multiplicity.
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| 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'''.
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| 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.
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| ==Rigorous statement==
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| Suppose that ''X'' and ''Y'' are two plane [[projective plane|projective]] curves defined over a [[field (mathematics)|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 [[intersection number|multiplicities]], is equal to the product of the degrees of ''X'' and ''Y''.
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| The generalization in higher dimension may be stated as:
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| Let ''n'' [[projective variety|projective hypersurface]]s be given in a [[projective space]] of dimension ''n'' over an algebraic closed field, which are defined by ''n'' [[homogeneous polynomial]]s in ''n'' + 1 variables, of degrees <math>d_1, \ldots,d_n.</math> Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product <math>d_1 \cdots d_n.</math> If the hypersurfaces are irreducible and in relative [[general position]], then there are <math>d_1, \ldots,d_n</math> intersection points, all with multiplicity 1.
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| 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 <math>\delta</math> and [[degree of an algebraic variety|degree]] <math>d_1</math>, and ''H'' is a hypersurface (defined by a polynomial) of degree <math>d_2</math>, that does not contain any [[irreducible component]] of ''V'', then the intersection of ''V'' and ''H'' has dimension <math>\delta-1</math> and degree <math>d_1d_2.</math> For a (sketched) proof using the [[Hilbert series]] see [[Hilbert series and Hilbert polynomial#Degree of a projective variety and Bézout's theorem]].
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| ==History==
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| Bezout's theorem was essentially stated by Isaac Newton in his proof of [[Newton's theorem about ovals|lemma 28]] of volume 1 of his ''[[Philosophiæ Naturalis Principia Mathematica|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.<ref>{{cite book | authorlink=Frances Kirwan | last=Kirwan | first=Frances | title=Complex Algebraic Curves | publisher=Cambridge University Press| location=United Kingdom | year=1992 | isbn=0-521-42353-8}}</ref>
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| ==Intersection multiplicity==
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| {{see|Intersection number}}
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| 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 line]]s 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.
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| ==Examples==
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| *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''+2''y''=3 and ''x''+2''y''=5 are represented by the homogeneous equations ''x''+2''y''-3''z''=0 and ''x''+2''y''-5''z''=0. Solving, we get ''x''= -2''y'' 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.
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| *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 - x''<sup>2</sup> = 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.
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| * Two [[conic section]]s 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:
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| **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
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| ::<math>(x-a)^2+(y-b)^2 = r^2</math>
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| :in [[homogeneous coordinates]], we get
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| ::<math>(x-az)^2+(y-bz)^2 - r^2z^2 = 0,</math>
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| :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.
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| **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.
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| **The following pictures show examples in which the circle ''x''<sup>2</sup>+''y''<sup>2</sup>-1=0 meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than 1:
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| [[File:dbldbl.png]]
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| ::<math>x^2+4y^2-1=0:\ \hbox{two intersections of multiplicity 2}</math>
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| [[File:intersect3.png]]
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| ::<math>5x^2+6xy+5y^2+6y-5=0:\ \hbox{an intersection of multiplicity 3}</math>
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| [[File:intersect4.png]]
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| ::<math>4x^2+y^2+6x+2=0:\ \hbox{an intersection of multiplicity 4}</math>
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| ==Sketch of proof==
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| Write the equations for ''X'' and ''Y'' in [[homogeneous coordinates]] as
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| :<math>a_0z^m + a_1z^{m-1} + \dots + a_{m-1}z + a_m = 0</math>
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| :<math>b_0z^n + b_1z^{n-1} + \dots + b_{n-1}z + b_n = 0</math>
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| where ''a<sub>i</sub>'' and ''b<sub>i</sub>'' 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
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| :<math>S=\begin{pmatrix}
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| a_0 & a_1 & a_2 & a_3 & a_4 & 0 & 0 \\
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| 0 & a_0 & a_1 & a_2 & a_3 & a_4 & 0 \\
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| 0 & 0 & a_0 & a_1 & a_2 & a_3 & a_4 \\
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| b_0 & b_1 & b_2 & b_3 & 0 & 0 & 0 \\
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| 0 & b_0 & b_1 & b_2 & b_3 & 0 & 0 \\
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| 0 & 0 & b_0 & b_1 & b_2 & b_3 & 0 \\
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| 0 & 0 & 0 & b_0 & b_1 & b_2 & b_3 \\
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| \end{pmatrix}.</math>
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| 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 (a<sub>0</sub>)<sup>n</sup>(b<sub>n</sub>)<sup>m</sup>, all have degree ''mn'', so |''S''| is a homogeneous polynomial of degree ''mn'' in ''x'' and ''y'' (recall that ''a''<sub>''i''</sub> and ''b''<sub>''i''</sub> 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.<ref>Follows [http://www.archive.org/details/cu31924001544216 ''Plane Algebraic Curves''] by Harold Hilton (Oxford 1920) p. 10</ref>
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| ==See also==
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| *[[AF+BG theorem]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite book | author=William Fulton | authorlink=William Fulton (mathematician) | title=Algebraic Curves | series=Mathematics Lecture Note Series | publisher=W.A. Benjamin | year=1974 | id={{Listed Invalid ISBN|0-8053-3081-4}} | page=112 }}
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| *{{citation| author-link = Isaac Newton|last=Newton |first=I.| year = 1966| title = [[Philosophiæ Naturalis Principia Mathematica|Principia Vol. I The Motion of Bodies]]| edition = based on Newton's 2nd edition (1713); translated by Andrew Motte (1729) and revised by [[Florian Cajori]] (1934)| publisher = University of California Press| location = Berkeley, CA| isbn = 978-0-520-00928-8}} Alternative translation of earlier (2nd) edition of Newton's ''Principia''.
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| * (generalization of theorem) http://mathoverflow.net/questions/42127/generalization-of-bezouts-theorem
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| ==External links==
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| * {{springer|title=Bezout theorem|id=p/b016000}}
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| * {{MathWorld|title=Bézout's Theorem|urlname=BezoutsTheorem}}
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| * [http://www.mathpages.com/home/kmath544/kmath544.htm Bezout's Theorem at MathPages]
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| {{DEFAULTSORT:Bezouts Theorem}}
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| [[Category:Theorems in plane geometry]]
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| [[Category:Incidence geometry]]
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| [[Category:Intersection theory]]
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| [[Category:Theorems in algebraic geometry]]
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Prostatic congestion and inflammation can stimulate the prostate urethra inducing frequent spermatorrhea. Increase Testosterone Level - For a long time it was thought that certain unwanted symptoms were just part of the aging process and that there was nothing that could be done to treat them. Treatments or medications used for prostrate cancer reduce the manufacture of testosterone in the male testes and adrenal glands. There are many reasons that business ethics matter. Most of these herbs have been in use for centuries, used by Africans, Asians, Americans etc.
Here are some remedies for impotence you can try tonight. Impotence (or erectile dysfunction) is defined as the inability to achieve or maintain an erection sufficient for sexual intercourse, and includes the inability to get an erection as a result of sexual stimulation or to lose your erection prior to ejaculation. In a published study reported in the March issue of Urology, researchers surveyed over 1,000 men who underwent radical prostatectomy (RP) between 1962 and 1997 (85. A few of the patterns truly flow straight down to the brim of the cap, although some have add-ons decorating the top like organizations or metallic studs, and sometimes several rhinestones. To stay healthy, it is critical to get peaceful rest for not less than 8 hrs on a daily basis.
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