Intereting Posts

Can permutating the digits of an irrational/transcendental number give any other such number?
natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$
Induction proof concerning a sum of binomial coefficients: $\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$
If $p\mid4n!-1$ then $p>n$
What can you say about a continuous function that is zero at all integer values?
Cantor set: Lebesgue measure and uncountability
When does variété mean manifold?
Information on “stronger form” of Dirichlet's Theorem on Arithmetic Progressions
Is there a Second-Order Axiomatization of ZF(C) which is categorical?
Holomorphic extension of a function to $\mathbb{C}^n$
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
Bounds on a sum of gcd's
Normal subgroups of $S_4$
A manifold such that its boundary is a deformation retract of the manifold itself.
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$

How many roots can there be of a polynomial $f \in K[x_1, x_2, \dots , x_n]$ where $K$ is a finite field and the maximum exponent of $x_i$ in any term is $m$ for all $i$, assuming not all coefficients are zero?

I found this related question, but I’m really interested in the worst case; I won’t have any “singularity condition”, necessarily.

Here’s an idea:

- Showing that an integral domain is a PID if it satisfies two conditions
- For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?
- Example for an ideal which is not flat (and explicit witness for this fact)
- generators of a transitive permutation group
- An arrow is monic in the category of G-Sets if and only if its monic the category of sets
- Can finite non-isomorphic groups of the same order have isomorphic endomorphism monoids?

View $f$ as a polynomial over $(K(x_2, \dots , x_n))[x_1]$, where $K(x_2, \dots , x_n)$ is the field of fractions of $K[x_2, \dots , x_n]$. At least one coefficient of $f$ when viewed this way is non-zero. Now $f$ has at most $m$ roots, since its single indeterminate, $x_1$, has exponent at most $m$. For each of these $m$ roots $y_{1,1}, y_{1,2}, \dots, y_{1,m}$ and any $x_2, \dots, x_n$, $f(y_{1,j}, x_2, \dots, x_n) = 0$. There are $m k^{n-1}$ tuples of this form, where $k$ is the size of $K$.

Additionally, for each $v \in K$, $f(v, x_2, \dots, x_n)$ is a polynomial in $n-1$ variables. For the $v$s that are not roots of $f$ when viewed as above, $f(v, x_2, \dots, x_n)$ has at least one non-zero coefficient. This sets up an inductive equation:

$$

c(n) \leq

\begin{cases}

m & n = 1\\

m k^{n-1} + k c(n-1) & n > 1

\end{cases}

$$

With solution

$$

c(n) \leq n m k^{n-1}

$$

On the other hand, let $f_i$ be a polynomial of degree $m$ in $K[x_i]$ with $m$ distinct roots. Then $\prod f_i$ has at least $n m (k-m)^{n-1}$ distinct roots.

Is this right? Can we get tighter bounds?

- Proving $\mathbb{Q} = \{f(\sqrt{2}): f(x) \in \mathbb{Q}\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$
- Extending an automorphism to the integral closure
- Points and maximal ideals in polynomial rings
- Construction of $p^n$ field
- Has $S$ infinitely many nilpotent elements?
- A question about the proof that $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic
- Kernel of a substitution map
- Examples of a monad in a monoid (i.e. category with one object)?
- Hensel's Lemma and Implicit Function Theorem
- Euclidean domain in which the quotient and remainder are always unique

It turns out that the upper bound is just a special case of the Schwartz-Zippel lemma.

- What is the intuitive meaning of the scalar curvature R?
- Is there a computer program that does diagram chases?
- Looking for an explanation (vector application with bearings problem)
- Subgroups of an infinite group with a given index
- For what conditions on sets $A$ and $B$ the statement $A – B = B – A$ holds?
- Is a contradiction enough to prove a set equality to $\varnothing$?
- How can you prove that the square root of two is irrational?
- Finite index subgroups of a virtually abelian group
- When does intersection of measure 0 implies interior-disjointness?
- order of elliptic curve $y^2 = x^3 – x$ defined over $F_p$, where $p \equiv 3 \mod{4}$
- Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$
- Showing $\mathbb{Z}_6$ is an injective module over itself
- Solve a second order DEQ using Euler's method in MATLAB
- Does there exist a linearly independent and dense subset?
- Derivative of determinant of a matrix