Are there specific tests to indicate whether you have polycythemia vera?

Partially. Pv is a clinical diagnosis that is somewhat complex. A large proportion of pv patients (but not all) have a positive jak-2 gene rearrangement test. There are other causes of polycythemia that need to be considered as well- treatment of pv is important, so you should see a hematologist to make the diagnosis and formulate a treatment plan.
Absolutely. Simple is first a CBC with a differential. Also ask for a peripheral blood smear. You are looking for excessively high levels of hemoglobin or hematocrit.