Viktor Gyökeres would not count as a homegrown player in the Premier League if he joins Arsenal.
The Premier League’s homegrown player rule requires a player to have been registered with an English or Welsh club for at least three seasons before their 21st birthday. Gyökeres, although he was at Brighton & Hove Albion from 2018 and had loan spells in England, did not spend three full seasons registered before turning 21, and much of his early career was abroad or on loan outside England. His permanent breakthrough came later at Coventry and Sporting CP, so he does not meet the homegrown criteria.
In short, despite some time in England, Gyökeres is considered a non-homegrown player for Premier League squad registration purposes