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A '''quasi-homogeneous polynomial''' is a [[polynomial]] which has a degenerate [[Newton polygon]]. This means that if
They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Have you heard about niche marketing and advertising. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. However, you may not be able to find a theme that is in sync with your business. Contact Infertility Clinic Providing One stop Fertility Solutions at:. Make sure you have the latest versions of all your plugins are updated. <br><br>Millions of individuals and organizations are now successfully using this tool throughout the world. I don't want that for my visitors and I'm quite sure they don't either. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. Change the entire appearance of you blog using themes with one click.  If you have any questions pertaining to where and the best ways to make use of [http://miu.tw/backup_plugin_468644 wordpress dropbox backup], you could contact us at our website. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.
: <math>f(x)=\sum_\alpha a_\alpha x^\alpha\text{, where }\alpha=(i_1,\dots,i_r)\in \mathbb{N}^r \text{, and } x^\alpha=x_1^{i_1} \cdots x_r^{i_r}</math>
is a polynomial, then there ''r'' integers <math>w_1, \ldots, w_r</math>, called '''weights''' of the variables such that the sum <math>w=w_1i_1+ \cdots + w_ri_r</math> is the same for all terms of ''f''. This sum is called the ''weight'' or the ''degree'' of the polynomial. In other words, the [[convex hull]] of the set <math>\{\alpha | a_\alpha\neq0\}</math> lies entirely on an affine hyperplane.
 
The term ''quasi-homogeneous'' comes form the fact that a polynomial ''f'' is quasi-homogeneous if and only if
:<math> f(\lambda^{w_1} x_1, \ldots, \lambda^{w_r} x_r)=\lambda^w f(x_1,\ldots, x_r)</math>
for every <math>\lambda</math>  in the field of the coefficients. A [[homogeneous polynomial]] is quasi-homogeneous for all weights equal to 1.
 
==Introduction==
Consider the polynomial <math>f(x,y)=5x^3y^3+xy^9-2y^{12}</math>. This one has no chance of being a [[homogeneous polynomial]]; however if instead of considering <math>f(\lambda x,\lambda y)</math> we use the pair <math>(\lambda^3,\lambda)</math> to test ''homogeneity'', then
 
: <math>f(\lambda^3 x,\lambda y)=5(\lambda^3x)^3(\lambda y)^3+(\lambda^3x)(\lambda y)^9-2(\lambda y)^{12}=\lambda^{12}f(x,y). \, </math>
 
We say that <math>f(x,y)</math> is a quasi-homogeneous polynomial of '''type'''
(3,1), because its three pairs (''i''<sub>1</sub>,''i''<sub>2</sub>) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation <math>3i_1+1i_2=12</math>. In particular, this says that the Newton polygon of <math>f(x,y)</math> lies in the affine space with equation <math>3x+y=12</math> inside <math>\mathbb{R}^2</math>.
 
The above equation is equivalent to this new one: <math>\tfrac{1}{4}x+\tfrac{1}{12}y=1</math>. Some authors<ref>J. Steenbrink (1977). ''Compositio Mathematica'', tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at [http://www.numdam.org/ Numdam])</ref> prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type (<math>\tfrac{1}{4},\tfrac{1}{12}</math>).
 
As noted above, a homogeneous polynomial <math>g(x,y)</math> of degree ''d'' is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation <math>1i_1+1i_2=d</math>.
 
==Definition==
Let <math>f(x)</math> be a polynomial in ''r'' variables <math>x=x_1\ldots x_r</math> with coefficients in a commutative ring ''R''. We express it as a finite sum
 
: <math>f(x)=\sum_{\alpha\in\mathbb{N}^r} a_\alpha x^\alpha, \alpha=(i_1,\ldots,i_r), a_\alpha\in \mathbb{R}.</math>
 
We say that ''f'' is '''quasi-homogeneous of type''' <math>\varphi=(\varphi_1,\ldots,\varphi_r)</math>, <math>\varphi_i\in\mathbb{N}</math> if there exists some <math>a\in\mathbb{R}</math> such that
 
: <math>\langle \alpha,\varphi \rangle = \sum_k^ri_k\varphi_k=a,</math>
 
whenever <math>a_\alpha\neq 0</math>.
 
==References==
<references/>
 
[[Category:Commutative algebra]]
[[Category:Algebraic geometry]]

Latest revision as of 22:59, 13 November 2014

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Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer.

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