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Let $P_1,\ldots, P_d, Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots, x_n]$ be homogenous polynomials of degree at most $r$. Assume that $P_1 \cdot P_2 \cdots P_{d-1} \cdot P_d \in \langle Q_1, \ldots, Q …

Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$. Assume that these polynomials are pairwise coprime. Denote $P:= f_1 \cdot f_2 \ldots \cdo …

Let $q_1$ and $q_2$ be two irreducible quadratic homogeneous polynomials in $\mathbb{C}[x_0, \ldots, x_n]$. Consider the ideal $\langle q_1, q_2 \rangle$. When this ideal is prime? I am interesti …

Let $S$ be a system of polynomial equations over $\mathbb{F}_q$. Assume that $S$ has a solution in $\overline{\mathbb{F}_q}$. Denote by $k$ the minimal number such that $S$ has $\mathbb{F}_{q^k}$- …

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$. Let $r_1, \ld …

Let $Q_1, \ldots, Q_k$ and $P_1,\ldots, P_m$ be irredicable homogenous polynomials in $\mathbb{C}[x_0,\ldots, x_n]$ such that $V(Q_1, \ldots, Q_k) \subseteq \cup_i V(P_i)$. Here $V$ is projective vari …

There are many results about irreducible polynomials over finite fields: we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, …

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$. Assume that for every $i$ and f …

We will assume that $f$ is irreducible (if $f$ is not irreducible then in fact the argument of Zach Teitler's answer works). Consider $M:= f \cap P_1$ (I mean the intersection of the zeros $f$ and $P …

Yes. 1) A quadratic homogenous polynomial $f$ (over $\mathbb{C}$) is irreducible iff $\text{rk}(f) \ge 3$. Here $\text{rk}(f)$ is the rank of $f$ as a quadratic form. Indeed, if $\text{rk}(f) < 3$ …

This is a generalization of this question. Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f_1, \ldots, f_s$ be a homogenous quadrati …

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$. I mean there is polynomial reduction $F$ such that for every boolean circu …