Restricted sumset

In additive number theory and combinatorics, a restricted sumset has the form

${\displaystyle S=\{a_{1}+\cdots +a_{n}:\ a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}\ \mathrm {and} \ P(a_{1},\ldots ,a_{n})\not =0\},}$

where ${\displaystyle A_{1},\ldots ,A_{n}}$ are finite nonempty subsets of a field F and ${\displaystyle P(x_{1},\ldots ,x_{n})}$ is a polynomial over F.

If ${\displaystyle P}$ is a constant non-zero function, for example ${\displaystyle P(x_{1},\ldots ,x_{n})=1}$ for any ${\displaystyle x_{1},\ldots ,x_{n}}$, then ${\displaystyle S}$ is the usual sumset ${\displaystyle A_{1}+\cdots +A_{n}}$ which is denoted by ${\displaystyle nA}$ if ${\displaystyle A_{1}=\cdots =A_{n}=A.}$

When

${\displaystyle P(x_{1},\ldots ,x_{n})=\prod _{1\leq i<j\leq n}(x_{j}-x_{i}),}$

S is written as ${\displaystyle A_{1}\dotplus \cdots \dotplus A_{n}}$ which is denoted by ${\displaystyle n^{\wedge }A}$ if ${\displaystyle A_{1}=\cdots =A_{n}=A.}$

Note that |S| > 0 if and only if there exist ${\displaystyle a_{1}\in A_{1},\ldots ,a_{n}\in A_{n}}$ with ${\displaystyle P(a_{1},\ldots ,a_{n})\not =0.}$