Intereting Posts

Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Exponential map of Beltrami-Klein model of hyperbolic geometry
“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?
Determinant of a Special Symmetric Matrix
Are these two predicate statements equivalent or not?
probability almost surely and expectation
Definition of hyperbolic lenght.
Bounded (from below) continuous local martingale is a supermartingale
Proof of uniform convergence and continuity
How can I find the surface area of a normal chicken egg?
“What if” math joke: the derivative of $\ln(x)^e$
Proving without Zorn's Lemma: additive group of the reals is isomorphic to the additive group of the complex numbers
Nested Interval Property implies Axiom of Completeness
Universal coefficient theorem with ring coefficients

Can a non-zero polynomial in one variable have infinitely many roots ?

Can a non-zero polynomial in one variable have uncountably many roots ?

**Motivation** : over $\mathbb Z/12\mathbb Z$, $X^2-4$ has 4 roots.

- The number of real roots of $1+x/1!+x^2/2!+x^3/3! + \cdots + x^6/6! =0$
- Find $\alpha^{2016} + \beta^{2016} + \alpha^{2014} + \beta^{2014} \over \alpha^{2015} + \beta^{2015}$ for zeroes of the polynomial $x^2+3x +1$
- On a system of equations with $x^{k} + y^{k} + z^{k}=3$ revisited
- Can a (rational coefficient) polynomial have roots with two radicals?
- Understanding Primitive Polynomials in GF(2)?
- For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

When it comes to polynomials with coefficents over an integral domain, the answer is clearly negative (in that case, a polynomial can’t have more roots than its degree).

What happens with a ring that has zero divisors ?

- Decomposition of polynomial into irreducible polynomials
- Cyclotomic polynomials explicitly solvable??
- On a system of equations with $x^{k} + y^{k} + z^{k}=3$ revisited
- Modified Hermite interpolation
- Galois group of a quartic
- Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
- Proof of a lower bound of the norm of an arbitrary monic polynomial
- Two polynomial problem
- Roots of $y=x^3+x^2-6x-7$
- Positive integer solutions of $a^3 + b^3 = c$

Let $R = \prod_{i \in I} (\mathbb{Z}/4\mathbb{Z})$ with elementwise addition and multiplication. Then $t \mapsto (2,2,…)t$ is a non-zero polynomial over $R$ with $2^{|I|}$ many zeros.

Here’s a very simple construction: let $R$ be the ring

$$ \mathbf{Z}[x_0, x_1, x_2, \ldots] / \langle x_0^2 + 1, x_1^2 + 1, x_2^2 + 1, \ldots \rangle $$

Then every $x_i$ is a root of the polynomial $t^2 + 1$ over $R$.

A more trimmed down example is the ring

$$ \mathbf{Z}[x,y] / \langle x^2, xy, y^2 \rangle $$

This is the ring of all polynomials of the form $a + bx + cy$ with integer coefficients, subject to the relations $x^2 = xy = y^2 = 0$. So multiplication is

$$ (a + bx + cy)(d + ex + fy) = ad + (ae+bd) x + (af+cd)y $$

Every number of the form $bx + cy$ is a root of the polynomial $t^2$.

Of course, this example only needed one variable, not two, but I think it’s more interesting with two.

Take $A = \prod_{n \geq 1} \mathbb{Z}/2^n\mathbb{Z}$ and consider the polynomial $f(x) = 2x$ over $A.$ It has infinitely many zeros.

- Square root of a number squared is equal to the absolute value of that number
- Rate of Fourier decay of indicator functions
- Jordan normal form and invertible matrix of generalized eigenvectors proof
- Simple examples of $3 \times 3$ rotation matrices
- Question about soluble and cyclic groups of order pq
- Splitting of prime ideals in algebraic extensions
- Computing the Expectation of the Square of a Random Variable: $ \text{E} $.
- Determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$
- Are there areas of mathematics (current or future) that cannot be formalized in set theory?
- infinite length of a curve
- How to estimate of coefficients of logistic model
- Find the probability that a geometric random variable $X$ is an even number
- Real life usage of Benford's Law
- Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
- A basic question on expectation of distribution composed random variables