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Let $\{F_i(x)\}$ be homogeneous polynomials of degree $d>1$ in $n>1$ variables. Suppose also that the $F_i$ have no common zeros besides $0$. Prove the following relation cannot be satisfied. $$\sum_{i=1}^n x_iF_i(x)=0.$$

Essentially, this asks if the universal hyperplane in $P^n$ admits a certain kind of regular section. I know a geometric answer and am looking for an algebraic argument. It was suggested to me that the notion of a regular sequence might be helpful, but it does not seem necessary that the $F_i$ form one.

This is a small part of an exercise in Harris's Algebraic Geometry: A First Course, chapter 4. We work over $\mathbb C$.

Potato
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    If $F_i$ s are as above, with no common non-trivial zeroes, then it is a regular sequence. – Mohan Jan 29 '16 at 02:00
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    @Mohan I see. Then one can use the Koszul complex to finish, I think. – Potato Jan 29 '16 at 02:10
  • I think that you can prove that if the relation is valid, then the intersection of their zero sets has codimension two. But my argument is also geometric... – Alan Muniz Jan 29 '16 at 15:21
  • Maybe you can extract something from the proof of Proposition 2.6 of Jouanolou's book Equations de Pfaff Algébriques – Alan Muniz Jan 29 '16 at 15:31

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The key words here are "zero-dimensional ideal".

In the following $F_i\in K[X_1,\dots,X_n]$, where $K$ is an algebraically closed field.

Since the system of polynomial equations $F_i(x)=0$, $1\le i\le n$, has only finitely many solutions, that is, the ideal $I=\langle F_1,\dots,F_n\rangle$ is contained in only finitely many maximal ideals, $I$ is a zero-dimensional ideal. Equivalently, $K[X_1,\dots,X_n]/I$ is artinian. This shows that $F_1,\dots,F_n$ is a homogeneous system of parameters, and since $K[X_1,\dots,X_n]$ is Cohen-Macaulay we get that $F_1,\dots,F_n$ is a regular sequence.

Now, from $\sum_{i=1}^nX_iF_i=0$ it follows $X_nF_n\in\langle F_1,\dots,F_{n-1}\rangle$, so $X_n\in\langle F_1,\dots,F_{n-1}\rangle$, a contradiction with $d>1$.

Remark. The claim holds for homogeneous polynomials $F_i$, $1\le i\le n$, with $\deg F_i=d_i\ge 2$ and having a finite zero locus.

user26857
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  • For more details on zero-dimensional ideals see this answer and the linked threads. – user26857 Jan 30 '16 at 19:10
  • Thank you. Is the following alternative argument also correct? Given that the polynomials form a regular sequence, the only possible relations among them are the Koszul relations (since the Koszul complex is exact). But the given identity is not one of those relations. – Potato Jan 30 '16 at 20:32
  • I think it is, but it's more complicated than necessary. – user26857 Jan 30 '16 at 21:05