Intereting Posts

Translation operator and continuity
If $a,b$ are integers such that $a \mid x$ and $b \mid x$ , must $\mathrm{lcm}(a,b)\mid x$?
L'Hôpital in several variables
Problem concerning Sylow subgroups and normalizers
dissection of rectangle into triangles of the same area
Equivalence of geometric and algebraic definitions of conic sections
What Is a Morphism?
Calculating a Lebesgue integral involving the Cantor Function
$\epsilon – \delta$ definition to prove that f is a continuous function.
Proving the Product Rule for exponents with the same base
Rational analysis
If $I = \langle 2\rangle$, why is $I$ not a maximal ideal of $\mathbb Z$, even though $I$ is a maximal ideal of $\mathbb Z$?
(Counting problem) very interesting Modular N algebraic eqs – for combinatorics-permutation experts
How to calculate the probability of rolling 6 at least 5 times in a row, out of 50 tries?
How many different sudoku puzzles are there?

Starting from this question, we set $n=k=2$ and use the function $f\in\Bbb Z[x,y]$ where $f(x,y)=x\cdot y+x+y$, then the proofs applied to that question satisfy this case.

Note that for $k=1$ the function $f(x)=x$ satisfies the title statement (except the $k$ condition) for all $n$, so this case is considered not to be part of this question.

Note further that if $n$ is prime, the title statement is easy to prove:

- How do you prove this very different method for evaluating $\sum_{k=1}^nk^p$
- Any Implications of Fermat's Last Theorem?
- Alternative proofs that $A_5$ is simple
- Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
- $p=4n+3$ never has a Decomposition into $2$ Squares, right?
- **Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

Let $n\ge 2$ and $k\ge 2$ and let $m$ be the multiplicative order$\mod n$. The function $f\in\Bbb Z[x_1,\dots,x_k],f(\overline x)=\prod\limits_{i=1}^k(x^m-1)-(-1)^k$ will have value $f(\overline x)\equiv (-1)^k\pmod n$ whenever $\exists i:$GCD$(x_i, n)=1$ (since $x_i^m-1\equiv 0\pmod n$ in that case), so the important part to consider is $\forall i:$GCD$(x_i,n)\gt 1$. For prime $n$, this will occur exactly when $\overline x\equiv \overline 0\pmod n$, so that case need not be considered further.

What proof is there for the title statement for $n\ge 2,k\ge 2$ and $n$ is squarefree? (Notation: $\overline x\in\Bbb Z^k$ is the “vector $x$” having dimension $k$; $\overline 0$ is the “zero vector” i.e., a vector of specified size having all entries be $0$.)

To prove:

- For all $k\ge 1$ and squarefree $n$ there exists a function $f\in\Bbb Z[x_1,\dots,x_k] : f(\overline x)\equiv 0\pmod n\iff \overline x\equiv \overline 0\pmod n$ (proven, but feel free to offer an alternate proof, especially along the lines of the contrapositive)
- For all $k\ge 2$ and squareful $n$ no such function exists

- Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction.
- Make this visual derivative of sine more rigorous
- I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?
- For all integers a, b, c, if a | b and b | c then a | c.
- Counting non-isomorphic relations
- Need a counter example for cycle in a graph
- For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$
- How many essential ways: $n$ balls in $k$ bins, where $k$ can vary and there are $\geq 2$ balls in each bin?
- Equivalences to “D-finite = finite”
- How to determine which amounts of postage can be formed by using just 4 cent and 11 cent stamps?

If $R$ has an ideal $I$ where the multiplication from $R$ is $0$ on $I$ (which is true for $Z/p^{t+2}Z$ by taking $I$ to be the elements divisible by $p^{t+1}$, and false for $Z/nZ$ with $n$ squarefree), no polynomial in $k \geq 2$ variables meets the conditions of the problem. Fix a nonzero element $j \in I$ and let $(x_1,\dots,x_k)$ run through the $k$-tuples that have all but one coordinate equal to $0$, and the other coordinate equal to $j$. For our polynomial to be nonzero on these $k$-tuples, its degree $ \leq 1$ part must be $\sum a_i x_i$ with every $ja_i \neq 0$. But then we can set $y_1 = a_2j$ and $y_2 = -a_1j$ and let $Y$ be the $k$-tuple $(y_1,y_2, 0,0,0, \dots, 0)$. We have found a nonzero vector with $f(Y)=0$.

If $R$ is a finite field of $q$ elements, the constructions used in proving Chevalley-Warning type theorems suffice to form the desired polynomial, such as $f=1 – \prod(1-x_i^{q-1})$.

The Chinese remainder theorem argument shows that for any ring $R$ of characteristic $n > 0$, there is a polynomial in $k$ variables vanishing only at $(0,0,0…,0)$ if and only if there is one over every $p$-power torsion part $R_p$. There is a finite direct sum decomposition $R = \oplus R_{p_i}$ and the isomorphisms in both directions are integer linear combinations, which apply equally well to polynomials (with $\mathbb{Z}$ or $R$ coefficients). The problem therefore reduces to the case where $p^uR=0$ for a prime power $p^u > 1$. I think that for finite rings the problem is always reduced to one of the two cases listed above; solutions exist for finite fields and no other finite ring of characteristic a power of $p$. Taking direct sums we get the general classification. [And if the “I think…” is incorrect then at least we have a proof for the original problem.]

With $n$ squarefree, the ring $\def\Z{\mathbf Z}\Z/n\Z$ is a product of fields $\Z/p\Z$ for the prime divisors $p$ of$~n$, by the Chinese remainder theorem, and $(\Z/n\Z)[X_1,\ldots,X_k]$ is the product of the corresponding polynomial rings $R_p=(\Z/p\Z)[X_1,\ldots,X_k]$. So it suffices to find for each such$~p$ an element $f_p\in R_p$ with $f_p[\overline a]=0\iff\overline a=\overline 0$ for all $\overline a\in(\Z/p\Z)^k$. Given these, one can construct the corresponding element $f_n\in(\Z/n\Z)[X_1,\ldots,X_k]$ such that for each monomial in the $X_i$s, the coefficient of$~f$ reduces modulo$~p$ to the corresponding coefficient of$~f_p$, for all$~p$; such a coefficient exists (uniquely) by the Chinese remainder theorem. For $\overline a\in(\Z/n\Z)^k$, the reduction modulo$~p$ of $f_n[\overline a]$ is $f_p[\overline a]$, and this is zero for all$~p$ if and only if $\overline a=\overline 0\in(\Z/n\Z)^k$. Any lift$~f$ of $f_n$ to $\Z[X_1,\ldots,X_k]$ will answer the question.

The polynomial $f_p=(p-1)!^k-\prod_{i=1}^k\prod_{a=1}^{p-1}(X_i-a)\in\Z[X_1,\ldots,X_k]$ has the property of the question for $n=p$ prime, since the product is designed to give a factor divisible by$~p$ in its evaluation at $\overline a$ if and only if $\overline a\not\equiv\overline0\pmod p$, and the constant that it is subtracted from is the value of the product at$~\overline0$. Its reduction modulo$~p$ gives a suitable element in $(\Z/p\Z)[X_1,\ldots,X_k]$ for the above.

- Bounded, non-constant harmonic functions: how far are they from existing?
- Knights and knaves: Who are B and C? (task 26 from “What Is the Name of This Book?”)
- Why do we require a topological space to be closed under finite intersection?
- Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$
- The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$
- The sum of three colinear rational points is equal to $O$
- What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
- Is the adjoint representation of SO(4) self-dual?
- Equivalent definitions of isometry
- Choosing the correct subsequence of events s.t. sum of probabilities of events diverge
- On the ring generated by an algebraic integer over the ring of rational integers
- Lindelöf'ize a space?
- How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x – 1}^x {f\left( t \right)\,dt} $?
- In a reduced ring the set of zero divisors equals the union of minimal prime ideals.
- What happens to the frequency-spectrum if this sine-signal gets reset periodically?