Generally speaking the home page can be called to ways: (1) in absence of {segment_1}
or (2) when the site is using the designated primary template_group and index.
Let's deal the first one first, you'd just use the conditional
{if segment_1 ==""} what you want {/if}
Will test for the home page. Now, if you also consider it the 'homepage' when people are on the designated 'index' page, regardless of if they are at your base domain, then you'd want
{if segment_1=="archive" && segment_2=="index" OR segment_1==""} {/if}
where 'archive' would be whichever template_group you set to hold the primary index page.
You'll want to test the second option, but it should work. You'd want to use the conditional that requires the AND statement before the condition that is the OR.