Difference between revisions of "Homogenization"
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'''Homogenizing''' is a useful technique to solve certain multivariate [[inequality|inequalities]]. Given an inequality of the form <math>P(a_1,a_2, \ldots, a_n) \ge 0</math>, where <math>P</math> 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. | '''Homogenizing''' is a useful technique to solve certain multivariate [[inequality|inequalities]]. Given an inequality of the form <math>P(a_1,a_2, \ldots, a_n) \ge 0</math>, where <math>P</math> 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 <math>a,b,c>0</math> and <math>a+b+c=1</math>, prove that <math>a^2+b^2+c^2+1\ge 4(ab+bc+ca)</math>. | ||
| + | ==Solution== | ||
| + | So all the terms except for the <math>1</math> are of the second degree. We substituting <math>a+b+c</math> for the <math>1</math> in the inequality gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get <math>a^2+b^2+c^2+2(ab+bc+ca)=1</math>. 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 Cauchy-Schwarz Inequality. | ||
| + | |||
| + | We can use homogenization to help us solve these types of problems, especially inequalities however it's use is not limited. After making something homogenous we can often apply well known inequalities to solve problems. | ||
{{stub}} | {{stub}} | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
Revision as of 20:51, 22 January 2014
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 theare of the second degree. We substituting
for the
. Now plugging this for
, which is well-known by Cauchy-Schwarz Inequality.
are of the second degree. We substituting 
for the 
. Now plugging this for 
, which is well-known by Cauchy-Schwarz Inequality.
We can use homogenization to help us solve these types of problems, especially inequalities however it's use is not limited. After making something homogenous we can often apply well known inequalities to solve problems.
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