mathematics

Quasi-homogeneous polynomial

Quasi-homogeneous polynomial is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quasi-homogeneous polynomial rather than just read about it. In short: In algebra, a multivariate polynomial f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , {\displaystyle 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}},} is quasi-homogeneous or weighted homogeneous, if there exist r integers w 1 , … , w r {\displaystyle w_{1}…

Key takeaways

Quasi-homogeneous polynomial belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quasi-homogeneous polynomial to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quasi-homogeneous polynomial from memory before moving on to harder problems.

Reference excerpt

In algebra, a multivariate polynomial

f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , {\displaystyle 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}},}

is quasi-homogeneous or weighted homogeneous, if there exist r integers w 1 , … , w r {\displaystyle w_{1},\ldots ,w_{r}} , called weights of the variables, such that the sum w = w 1 i 1 + ⋯ + w r i r {\displaystyle w=w_{1}i_{1}+\cdots +w_{r}i_{r}} is the same for all nonzero terms of f. This sum w is the weight or the degree of the polynomial. The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if

f ( λ w 1 x 1 , … , λ w r x r ) = λ w f ( x 1 , … , x r ) {\displaystyle f(\lambda ^{w_{1}}x_{1},\ldots ,\lambda ^{w_{r}}x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})}

for every λ {\displaystyle \lambda } in any field containing the coefficients. A polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} is quasi-homogeneous with weights w 1 , … , w r {\displaystyle w_{1},\ldots ,w_{r}} if and only if

f ( y 1 w 1 , … , y n w n ) {\displaystyle f(y_{1}^{w_{1}},\ldots ,y_{n}^{w_{n}})}

is a homogeneous polynomial in the y i {\displaystyle y_{i}} . In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1. A polynomial is quasi-homogeneous if and only if all the α {\displaystyle \alpha } belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set { α ∣ a α ≠ 0 } , {\displaystyle \{\alpha \mid a_{\alpha }\neq 0\},} the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polytope (here "degenerate" means "contained in some affine hyperplane").

Introduction Consider the polynomial f ( x , y ) = 5 x 3 y 3 + x y 9 − 2 y 12 {\displaystyle f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{12}} , which is not homogeneous. However, if instead of considering f ( λ x , λ y ) {\displaystyle f(\lambda x,\lambda y)} we use the pair ( λ 3 , λ ) {\displaystyle (\lambda ^{3},\lambda )} to test homogeneity, then

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Quasi-homogeneous polynomial

Start with the simplest possible case. Write down what Quasi-homogeneous polynomial claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quasi-homogeneous polynomial before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quasi-homogeneous polynomial ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quasi-homogeneous polynomial

In research: Quasi-homogeneous polynomial appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quasi-homogeneous polynomial in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quasi-homogeneous polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Commutative algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Quasi-homogeneous polynomial outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Quasi-homogeneous polynomial in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quasi-homogeneous polynomial means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quasi-homogeneous polynomial out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasi-homogeneous polynomial in simple terms?

In algebra, a multivariate polynomial f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , {\displaystyle 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_…

Why does Quasi-homogeneous polynomial matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quasi-homogeneous polynomial?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quasi-homogeneous polynomial.