veldt is an older spelling of Dutch and Afrikaans veld "unenclosed country or open pasture-land", cognate with English field, from Proto-Germanic *felþa- from Proto-Indo-European *pelh₂- "flat, to spread".
In Latin *pelh₂- became plānus "flat, level, even, plain, clear" and Italian piano "quiet". English piano is a shortening of pianoforte, so named because the instrument could produce both loud and soft sounds.
polka is the feminine form of Polish polák "Pole", but further etymology is uncertain.
In Slavic, *pelh₂- became Russian полый polyj "open, free, uncovered" and Czech pole "field". The AHD connects this with polka, but the OED says this is discredited. Another discredited theory is that it is from Czech půlka "half", as in "half-step" or somesuch.