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A quartic is a polynomial of degree of a polynomial four. Its leading coefficient is nonzero.

A quasipolynomial is a function $f:\mathbb{Z}\to \mathbb{C}$ for which there are polynomials $P_0,\dots ,P_{m-1}$ such that $f(n)=P_r(n)$ whenever $n\equiv r(\mathrm{mod}m)$. The least possible positive $m$ is its period.

In polynomial division $q=Qs+r$ over a field, the polynomial quotient is the polynomial $s$. The polynomial remainder $r$ has smaller polynomial degree than the nonzero divisor $Q$ unless it is zero.

The ramification polynomial records the normalised differences of conjugates of a uniformizer, including the zero difference from the identity automorphism. Its quotient by $X$ is the reduced ramification polynomial. In the usual coefficient-index convention, the finite slopes of its lower Newton polygon are negatives of the valuations of its nonzero roots; the coefficient at the zero root is treated as having infinite valuation.

Split a rational $S{L}_m$ representation by its finite scalar center: on $W_k$, $\rho (\zeta I)={\zeta }^{k}I$ for ${\zeta }^m=1$. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of $\rho {|}_{W_k}$ can be represented on $S{L}_m$ by a sum of homogeneous polynomials $P_d$ with $d\equiv k(\mathrm{mod}m)$, by averaging over the scalar center. Its extension is ${\sum }_d(\det g{)}^{(k-d)/m}{P}_d(g)$, a regular function on $G{L}_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

A rational function is a quotient $p/q$ of polynomials, with $q$ nonzero. After cancelling common factors, it defines a holomorphic map from the Riemann sphere to itself.

A rational representation is a group homomorphism whose matrix coefficients are regular functions on the affine algebraic group. For the general linear group, those functions are polynomials in matrix entries and the inverse determinant. The word rational does not permit poles at points of the group.

For a dominant integer tuple $\lambda$, choose $k=\max (0,-{\lambda }_m)$ and define $D_{\lambda }={\det }^{-k}\otimes D_{\lambda +k(1,\dots ,1)}$, using the polynomial Schur module on the right. The alternant character formula for the general linear group shows that a larger admissible shift yields the same irreducible representation.

The minimal polynomial of a real matrix, initially computed over the complex numbers, has real coefficients. Its coefficientwise complex conjugate is also a monic polynomial of the same degree annihilating the matrix. Uniqueness of the minimal polynomial forces the two polynomials to agree.

For a nonzero real-on-the-circle trigonometric polynomial of actual order $d$, the polynomial $P(z)=z^{d}p(z)$ satisfies $P(z)=z^{2d}\overline{P(1/\overline{z})}$. Its constant term is the nonzero conjugate of its leading coefficient. Hence its nonzero roots of a polynomial occur in pairs $\zeta ,1/\overline{\zeta }$ with the same multiplicity of a root. The coefficient reversal and complex conjugation preserve the exponents of the paired factors. Fixed roots of a polynomial on the unit circle require nonnegativity, rather than this symmetry alone, to have even multiplicity.

Let $P$ be a monic polynomial of positive degree $m\ge 1$ with distinct roots of a polynomial $r_j$. The partial fraction decompositionfollows by matching the simple pole residues; the difference is entire and vanishes at infinity. Comparing its expansion there gives ${\sum }_j{r}_j^{\ell }/P^{\prime }(r_j)=0$ for $0\le \ell <m-1$ and ${\sum }_j{r}_j^{m-1}/P^{\prime }(r_j)=1$. These identities give the continuity and derivative jumps needed for constant-coefficient Green functions.

For a degree-$n$ totally ramified Galois extension and a uniformizer $\pi$, let $f$ be its minimal polynomial. Dividing the ramification polynomial $f(\pi (1+X))/{\pi }^n$ by its known factor $X$ gives the reduced ramification polynomial. It is monic of degree $n-1$, integral over the extension's valuation ring, and has roots $(\sigma (\pi )-\pi )/\pi$ for nonidentity automorphisms. Its constant coefficient is nonzero in a separable extension. Removing the zero root allows the usual Newton polygon root valuation theorem to apply directly.

For distinct elements $a_1,\dots ,a_n$ of a finite field, a Reed-Solomon code consists of the evaluation vectors $(f(a_1),\dots ,f(a_n))$ of polynomials with degree less than $k$. It has dimension $k$ and minimum Hamming distance $n-k+1$, so it is a maximum distance separable code. A nonzero polynomial has at most its degree many distinct polynomial roots. Primitive cyclic Reed-Solomon codes use all nonzero field elements, ordered as powers of a primitive root of unity; arbitrary evaluation sets need not give a cyclic code.

For a degree-$d$ polynomial, reflecting the usual coefficient-index axis replaces $(j,v(a_j))$ by $(d-j,v(a_j))$. The reflected lower polygon has slopes equal to root valuations, rather than their negatives. Horizontal lengths, and therefore multiplicities, are unchanged. This convention makes positive lower ramification groups breaks appear as positive slopes of a reduced ramification polygon.

The polynomial remainder is the polynomial $r$ in polynomial division $q=Qs+r$. Division by $X-a$ has a constant polynomial remainder $q(a)$, as expressed by the factor theorem.

Suppose every point of ${\mathbb{F}}_q^n$ lies on an affine line in a vector space meeting a subset $N$ in at least $m$ points, with $1\le m\le q$. A smaller $N$ would admit a nonzero multivariate polynomial of total degree at most $m-1$ vanishing on it, by the dimension of a bounded-total-degree polynomial space. Each selected line then has more roots of a polynomial than the degree of its restriction, so the polynomial vanishes at every point. The Schwartz-Zippel lemma forbids this because $m-1<q$. Thus the displayed bound holds, and is at least $m^n/n!$. This is a point-covering condition; the finite-field Kakeya set condition instead quantifies over directions.

Since $2\not\equiv 1(\mathrm{mod}4)$, the ring of integers of $\mathbb{Q}(\sqrt{2})$ is $\mathbb{Z}[\sqrt{2}]$, with integral basis $1,\sqrt{2}$ and field discriminant eight.

If a nonzero real polynomial has $N$ roots in an open real interval, counted with multiplicity of a root, its derivative has at least $N-1$ roots in that interval, also counted with multiplicity, provided $N\ge 1$. Each root of multiplicity $m$ becomes a root of the derivative of multiplicity $m-1$. The Rolle theorem supplies one further root between each pair of distinct consecutive roots. Thus $k$ distinct roots of multiplicities $m_j$ give at least ${\sum }_j(m_j-1)+(k-1)=N-1$ derivative roots.

Rolle's theorem is the equal-endpoint case of the mean value theorem. Let $a<b$ and let the real-valued function $f:[a,b]\to \mathbb{R}$ be continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. If $f(a)=f(b)$, there exists $c\in (a,b)$ with $f^{\prime }(c)=0$. Geometrically, a tangent line somewhere in the interior is horizontal, like the secant line joining the endpoints. ^[ref]

For the polynomial $f(x)=x(x-2)$ on $[0,2]$, both endpoint values are zero and the derivative is $f^{\prime }(x)=2x-2$. Thus $c=1$, where $f(1)=-1$, and the horizontal tangent line is $y=-1$. The horizontal secant line is $y=0$: equal endpoint values do not force the interior tangent line to pass through those endpoints.

The absolute value function $f(x)=|x|$ on $[-1,1]$ exposes the importance of differentiability. It is continuous and $f(-1)=f(1)=1$, so the secant line has slope zero. However,and the derivative does not exist at zero because the one-sided derivatives disagree. Every existing derivative in the interior is nonzero. The corner fails a hypothesis of both Rolle's theorem and the mean value theorem, so neither conclusion is guaranteed.

For a short proof, the extreme value theorem supplies a greatest and a least value on $[a,b]$. If $f$ is a constant function, every interior point works. Otherwise at least one of these values differs from the common endpoint value, so it is attained at an interior extremum. At an interior local maximum or local minimum, differentiability forces the derivative to be zero: the difference quotients from the two sides have opposite weak signs and the same limit.

This turns repeated values into information about derivatives. Two distinct roots of a polynomial force a root of its derivative between them; iterating this idea gives the Rolle root count with multiplicities. More generally, a differentiable function with nonzero derivative throughout a real interval cannot repeat a value there, hence is injective. The beauty is that knowing only two equal values guarantees an exact local feature somewhere between them.