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Homogeneous polynomial
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<h2 id="properties">Properties</h2> <p>A homogeneous polynomial defines a <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous function</a>. This means that, if a <a href="/facts/Multivariate_polynomial/Lzak8VVx">multivariate polynomial</a> <i>P</i> is homogeneous of degree <i>d</i>, then </p> P ( λ x 1 , … , λ x n ) = λ d P ( x 1 , … , x n ) , {\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,} <p>for every λ {\displaystyle \lambda } in any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> containing the <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> of <i>P</i>. Conversely, if the above relation is true for infinitely many λ {\displaystyle \lambda } then the polynomial is homogeneous of degree <i>d</i>. </p><p>In particular, if <i>P</i> is homogeneous then </p> P ( x 1 , … , x n ) = 0 ⇒ P ( λ x 1 , … , λ x n ) = 0 , {\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,} <p>for every λ . {\displaystyle \lambda .} This property is fundamental in the definition of a <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a>. </p><p>Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial. </p><p>Given a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> (or, more generally, a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>) <i>K</i>, the homogeneous polynomials of degree <i>d</i> form a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> (or a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>), commonly denoted R d . {\displaystyle R_{d}.} The above unique decomposition means that R {\displaystyle R} is the <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a> of the R d {\displaystyle R_{d}} (sum over all <a href="/facts/Nonnegative_integer/u65og8JE">nonnegative integers</a>). </p><p>The dimension of the vector space (or <a href="/facts/Free_module/o6rC6yoJ">free module</a>) R d {\displaystyle R_{d}} is the number of different monomials of degree <i>d</i> in <i>n</i> variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree <i>d</i> in <i>n</i> variables). It is equal to the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> </p> ( d + n − 1 n − 1 ) = ( d + n − 1 d ) = ( d + n − 1 ) ! d ! ( n − 1 ) ! . {\displaystyle {\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.} <p> Homogeneous polynomial satisfy <a href="/facts/Euler%27s_homogeneous_function_theorem/wF9XX7s5">Euler's identity for homogeneous functions</a>. That is, if <i>P</i> is a homogeneous polynomial of degree <i>d</i> in the indeterminates x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} one has, whichever is the <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> of the coefficients, </p> d P = ∑ i = 1 n x i ∂ P ∂ x i , {\displaystyle dP=\sum _{i=1}^{n}x_{i}{\frac {\partial P}{\partial x_{i}}},} <p>where ∂ P ∂ x i {\displaystyle \textstyle {\frac {\partial P}{\partial x_{i}}}} denotes the <a href="/facts/Formal_derivative/actYz19a">formal partial derivative</a> of <i>P</i> with respect to x i . {\displaystyle x_{i}.} </p> <h2 id="homogenization">Homogenization</h2> <p>A non-homogeneous polynomial <i>P</i>(<i>x</i>1,...,<i>x</i><i>n</i>) can be homogenized by introducing an additional variable <i>x</i>0 and defining the homogeneous polynomial sometimes denoted <i>h</i><i>P</i>:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> h P ( x 0 , x 1 , … , x n ) = x 0 d P ( x 1 x 0 , … , x n x 0 ) , {\displaystyle {^{h}\!P}(x_{0},x_{1},\dots ,x_{n})=x_{0}^{d}P\left({\frac {x_{1}}{x_{0}}},\dots ,{\frac {x_{n}}{x_{0}}}\right),} <p>where <i>d</i> is the <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> of <i>P</i>. For example, if </p> P ( x 1 , x 2 , x 3 ) = x 3 3 + x 1 x 2 + 7 , {\displaystyle P(x_{1},x_{2},x_{3})=x_{3}^{3}+x_{1}x_{2}+7,} <p>then </p> h P ( x 0 , x 1 , x 2 , x 3 ) = x 3 3 + x 0 x 1 x 2 + 7 x 0 3 . {\displaystyle ^{h}\!P(x_{0},x_{1},x_{2},x_{3})=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.} <p>A homogenized polynomial can be dehomogenized by setting the additional variable <i>x</i>0 = 1. That is </p> P ( x 1 , … , x n ) = h P ( 1 , x 1 , … , x n ) . {\displaystyle P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Multi-homogeneous_polynomial/Jq3XWEMQ">Multi-homogeneous polynomial</a></li> <li><a href="/facts/Quasi-homogeneous_polynomial/yMXr4277">Quasi-homogeneous polynomial</a></li> <li><a href="/facts/Diagonal_form/95WoNKjl">Diagonal form</a></li> <li><a href="/facts/Graded_algebra/AIUjkSaP">Graded algebra</a></li> <li><a href="/facts/Hilbert_series_and_Hilbert_polynomial/FtFu54gb">Hilbert series and Hilbert polynomial</a></li> <li><a href="/facts/Multilinear_form/Rk420AcK">Multilinear form</a></li> <li><a href="/facts/Multilinear_map/YosSfBKE">Multilinear map</a></li> <li><a href="/facts/Polarization_of_an_algebraic_form/ysDI99Eo">Polarization of an algebraic form</a></li> <li><a href="/facts/Schur_polynomial/mBzF2Mdr">Schur polynomial</a></li> <li><a href="/facts/Symbol_of_a_differential_operator/E80ugWJb">Symbol of a differential operator</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li> Media related to Homogeneous polynomials at Wikimedia Commons</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/HomogeneousPolynomial.html">"Homogeneous Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9. <a href="978-0-387-20733-9" target="_blank">978-0-387-20733-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms homogeneous polynomial and form are sometimes considered as synonymous. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Linear forms are defined only for finite-dimensional vector space, and have thus to be distinguished from linear functionals, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces. <a href="/wiki/Linear_functional" target="_blank">/wiki/Linear_functional</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems. <a href="/wiki/Dimensional_analysis" target="_blank">/wiki/Dimensional_analysis</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Cox, Little & O'Shea 2005, p. 35 - Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9. <a href="https://books.google.com/books?id=QFFpepgQgT0C&pg=PP1" target="_blank">https://books.google.com/books?id=QFFpepgQgT0C&pg=PP1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>