Finite homogeneous maps and a sharp fiber bound

Definitions and mathematical statement

Let $m,n\geq1$, and let $f_1,\ldots,f_n\in{\mathbb{C}}[x_1,\ldots,x_m]$ be homogeneous of the same positive degree $d$, with \[ V(f_1,\ldots,f_n)=\{(0,\ldots,0)\}. \] Question 36 (Jarden) asks to prove that for every $\bar b=(b_1,\ldots,b_n)\in{\mathbb{C}}^n$ the affine fiber \[ \{\bar x\in{\mathbb{C}}^m:f_i(\bar x)=b_i\text{ for }1\leq i\leq n\} \] is finite, including the possibility of an empty fiber. Positive degree and positive numbers of variables and forms are the conventional nontrivial setting; the source does not explicitly list these integer bounds.

The source already includes its qualitative solution: homogenize to $f_i(x_1,\ldots,x_m)-b_i x_0^d=0$ in ${\mathbb{P}}^m$. An infinite affine fiber would yield a positive-dimensional projective closure meeting $x_0=0$, giving a common nonzero projective zero of the homogeneous forms, contrary to the hypothesis. This supplied solution is part of the record, not a newly proposed solution.

The aid states a stronger characteristic-free refinement. For any algebraically closed field $k$ and forms satisfying the analogous hypotheses, put $R=k[x_1,\ldots,x_m]$, $I=(f_1,\ldots,f_n)$, and \[ F:{\mathbb{A}}_k^m\longrightarrow{\mathbb{A}}_k^n,\qquad\bar x\longmapsto(f_1(\bar x),\ldots,f_n(\bar x)). \] Then $F$ is a finite morphism, and, for every $\bar b\in k^n$, \[ \operatorname{length}_k F^{-1}(\bar b) =\dim_k R/(f_1-b_1,\ldots,f_n-b_n) \leq\dim_k R/I\leq d^m. \] Thus the number of distinct points in every fiber is at most $d^m$, while the length counts multiplicity. Finiteness of the morphism does not assert surjectivity onto ${\mathbb{A}}_k^n$. The aid says the scheme-length bound is sharp in every characteristic, as illustrated by $n=m$, $f_i=x_i^d$ and the fiber over zero; it says the distinct-point bound is sharp when $\operatorname{char}(k)\nmid d$, using a fiber with all coordinates nonzero in the same example. No claim of reduced fibers in other characteristics is made.

Short English statement

Homogeneous forms with only the origin as a common zero have finite fibers over every target value. The source includes a projective proof. The supplied refinement makes the polynomial map finite and bounds every fiber's scheme length, hence its number of points, by $d^m$, with the stated sharpness qualifications.

Sources

• Local UnsolvedMath registry, permanent ID $20002372$, catalogue entry; aim-logic-notes.json, notes[147].
• American Institute of Mathematics, Extensions of Hilbert's tenth problem, Question 36 (Jarden), including the embedded projective-homogenization solution: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf. Supplied verification reports correction of subscripts, vector notation, and the run-together "C.Solution" extraction.
• Research attempt 1: full solution, problem solved at run, rigor label "proved," novelty confidence low. Registry classification solved in literature, status solved, statement status corrected/verified. No separate authoritative reference is supplied for the quantitative scheme-length refinement.