Homogenisation in inequalities

Can someone help me understand homogenisation and constraints application in inequalities?

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Calvin Lin @Surya Prakash @Aditya Raut @Mark Henning @Sharky Kesa

Abhi Kumbale - 4 years, 5 months ago

Sorry for being unable to answer due to the fact I have no in depth knowledge of inequalities.
But, I advise you to read Calvin Sir's note on 'How to ask for help?'; unable to provide link due to an error. It is in no way to offend you and I apologise if it does.

Yatin Khanna - 4 years, 5 months ago

What do you currently understand about it?

Calvin Lin Staff - 4 years, 5 months ago

I know it is based on the property that f (x,y,z)=t^(-n)f (tx,ty,tz). So any inequality can be written as a function of several variables and this property can be applied if the function is homogeneous. I would like to know it's various applications.

Abhi Kumbale - 4 years, 5 months ago

The basic idea is that a homogeneous polynomial (IE polynomial whose terms all have the same degree) can be easier to deal with than a polynomial whose terms have different degrees. As such, if we're given a condition, we try and use that to make the terms equal.

For example, a common condition is of the form $x + y + z = 1$ (or some other constant). If we had to deal with the expression $xy+yz+zx - xyz$ , we can homogenize by multplying the degree 2 terms by $x+y+z = 1$ , giving us

$xy+yz+zx - xyz \\ = (x+y+z)(xy+yz+zx) - xyz \\ = (x+y)(y+z)(z+x) \\ = (1-x)(1-y)(1-z) \\ \leq \left( \frac{2}{3} \right)^3$

This is a (constructed) example where the homogenization leads to a nice factoring, which leads to a nice way to deal with the expression.

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin – I have seen problems where no such constraints as above is given but one assumes such a constraint then solves the problem. How is this possible?

Abhi Kumbale - 4 years, 5 months ago

@Abhi Kumbale – In those cases, an assumption like $x + y + z = t$ is made, and then we introduce another variable.

In a sense, it is similar to "Richard Feymanns favourite integration trick of differentiating through the integral" (assuming you are familiar with how that works).

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin – I understand that you can introduce another variable but I have seen places where x+y+z=1 is introduced. Like this solution

https://brilliant.org/profile/surya-ou9y8q/sets/olympiad-proof-problems/354510/olympiad-proof-problem-day-1/

How is this right? I have also seen a similar approach in Engel's book.

Abhi Kumbale - 4 years, 5 months ago

@Abhi Kumbale – The key phrase is

The given expression is homogeneous equation of degree $0$ . So, we can make a constraint $a+b+c+d=1$ .

Do you understand why that means?

Suppose that $a + b + c + d = t$ , then consider the change of variables $A = \frac{a}{t} ,$ etc. What is $A + B + C + D$ ?

Why does "homogenous equation of degree 0" allow us to do this substitution? What happens if we have a "homogenous equation of degree 1"? What happens for a general polynomial say $a^2b + cd + c + 2$ ?

Calvin Lin Staff - 4 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$