Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let v be the number of distinct values of the polynomial f ( x ) = x 4 + a x 2 + b x , where a and b are elements of the finite field of size q, where q is odd. When b is 0, an exact formula for v can be given. When b is not 0, v = ( 5 / 8 ) q + O ( q ) , where the error term comes from the Riemann hypothesis. In this note  establish for the case that b is not 0, the inequality v ? ( q + 1 ) / 2 , without relying on the Riemann hypothesis.
['Robert C. Valentini']
I am writing a research paper on a idea A note on value sets of quartic polynomials
[{"from": "human", "value": "I am writing a research paper on a idea A note on value sets of quartic polynomials"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let v be the number of distinct values of the polynomial f ( x ) = x 4 + a x 2 + b x , where a and b are elements of the finite field of size q, where q is odd. When b is 0, an exact formula for v can be given. When b is not 0, v = ( 5 / 8 ) q + O ( q ) , where the error term comes from the Riemann hypothesis. In this note  establish for the case that b is not 0, the inequality v ? ( q + 1 ) / 2 , without relying on the Riemann hypothesis."}]