On value sets of polynomials over a field

Abstract

Let F be any field. Let p ( F ) be the characteristic of F if F is not of characteristic zero, and let p ( F ) = + ∞ otherwise. Let A 1 , … , A n be finite nonempty subsets of F, and let f ( x 1 , … , x n ) = a 1 x 1 k + ⋯ + a n x n k + g ( x 1 , … , x n ) ∈ F [ x 1 , … , x n ] with k ∈ { 1 , 2 , 3 , … } , a 1 , … , a n ∈ F ∖ { 0 } and deg g < k . We show that | { f ( x 1 , … , x n ) : x 1 ∈ A 1 , … , x n ∈ A n } | ⩾ min { p ( F ) , ∑ i = 1 n ⌊ | A i | − 1 k ⌋ + 1 } . When k ⩾ n and | A i | ⩾ i for i = 1 , … , n , we also have | { f ( x 1 , … , x n ) : x 1 ∈ A 1 , … , x n ∈ A n , and x i ≠ x j if i ≠ j } | ⩾ min { p ( F ) , ∑ i = 1 n ⌊ | A i | − i k ⌋ + 1 } ; consequently, if n ⩾ k then for any finite subset A of F we have | { f ( x 1 , … , x n ) : x 1 , … , x n ∈ A , and x i ≠ x j if i ≠ j } | ⩾ min { p ( F ) , | A | − n + 1 } . In the case n > k , we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.