A new extension of the Erdős–Heilbronn conjecture

Let A 1 , … , A n be finite subsets of a field F, and let f ( x 1 , … , x n ) = x 1 k + ⋯ + x n k + g ( x 1 , … , x n ) ∈ F [ x 1 , … , x n ] with deg g < k . We obtain a lower bound for the cardinality of { f ( x 1 , … , x n ) : x 1 ∈ A 1 , … , x n ∈ A n , and x i ≠ x j if i ≠ j } . The result extends the Erdős–Heilbronn conjecture in a new way.