Difference between revisions of "Homogenization"
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So all the terms except for the <math>1</math> are of the second degree. We substitute <math>a+b+c</math> for <math>1</math>. The inequality still gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get <cmath>a^2+b^2+c^2+2(ab+bc+ca)=1</cmath> Now plugging this for <math>1</math> in the inequality and simplifying gives <math>a^2+b^2+c^2\ge ab+bc+ca</math>, which is well-known by | So all the terms except for the <math>1</math> are of the second degree. We substitute <math>a+b+c</math> for <math>1</math>. The inequality still gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get <cmath>a^2+b^2+c^2+2(ab+bc+ca)=1</cmath> Now plugging this for <math>1</math> in the inequality and simplifying gives <math>a^2+b^2+c^2\ge ab+bc+ca</math>, which is well-known by | ||
− | the Rearrangement Inequality. | + | the [[Rearrangement Inequality]]. |
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− | [[Category: | + | [[Category:Algebra]] |
+ | [[Category:Inequalities]] | ||
+ | [[Category:Definition]] |
Latest revision as of 21:52, 6 February 2023
Homogenizing is a useful technique to solve certain multivariate inequalities. Given an inequality of the form , where
is a homogenous polynomial (that is, the degree of any term in the polynomial is the same), then we can arbitrarily impose a restraint of one order.
Example
If and
, prove that
.
Solution
So all the terms except for the are of the second degree. We substitute
for
. The inequality still gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get
Now plugging this for
in the inequality and simplifying gives
, which is well-known by
the Rearrangement Inequality.
We can use homogenization to help us solve these types of problems, especially inequalities. It's use is not limited. After making something homogenous we can often apply well known inequalities to solve problems.
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