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edit approved Jun 1, 2023 at 4:53
user347489
edit approved Jun 1, 2023 at 4:53
user347489
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Cubic polynomial over z_p$\mathbb{Z}_p$

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$ f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b, $$ where $$u\in\mathbb{Z}_p^*$$$u\in\mathbb{Z}_p^*$ is fixed  . Let Let $$S$$$S$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$$(a,b)\in\mathbb{Z}_p^2$ such that $$f_{a,b}(x)$$$f_{a,b}(x)$ factor linearly.Then what Then what is the cardinality of S$S$? Is it possible to get an exact formula somehow ?

Cubic polynomial over z_p

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$u\in\mathbb{Z}_p^*$$ is fixed  . Let $$S$$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$ such that $$f_{a,b}(x)$$ factor linearly.Then what is the cardinality of S? Is it possible to get an exact formula somehow ?

Cubic polynomial over $\mathbb{Z}_p$

Let $$ f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b, $$ where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all pairs $(a,b)\in\mathbb{Z}_p^2$ such that $f_{a,b}(x)$ factor linearly. Then what is the cardinality of $S$? Is it possible to get an exact formula somehow ?

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user126352
user126352

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$u\in\mathbb{Z}_p^*$$ is fixed . Let $$S$$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$ such that $$f_{a,b}(x)$$ factor linearly.Then what is the cardinality of S? Is it possible to get an exact formula somehow ?

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$u\in\mathbb{Z}_p^*$$ is fixed . Let $$S$$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$ such that $$f_{a,b}(x)$$ factor linearly.Then what is the cardinality of S?

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$u\in\mathbb{Z}_p^*$$ is fixed . Let $$S$$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$ such that $$f_{a,b}(x)$$ factor linearly.Then what is the cardinality of S? Is it possible to get an exact formula somehow ?

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user126352
user126352

Cubic polynomial over z_p

Let $$f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b.$$ where $$u\in\mathbb{Z}_p^*$$ is fixed . Let $$S$$ be the set consisting of all pairs $$(a,b)\in\mathbb{Z}_p^2$$ such that $$f_{a,b}(x)$$ factor linearly.Then what is the cardinality of S?