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Let $F$ be an entire function of one variable satisfying $|F(\zeta )|\le C(1+|\zeta |{)}^N{e}^{R|\mathrm{Im}\zeta |}$, and let $P$ be a nonzero polynomial. If $F/P$ is entire, it satisfies a bound of the same exponential type, with a possibly different polynomial power. Indeed, outside a sufficiently large disk, $|P(\zeta )|\ge c(1+|\zeta |{)}^{\mathrm{deg}P}$; inside the disk the extended quotient is bounded. Combining these estimates proves the claim. The same reasoning works with the exact support function of a compact interval. Zeros of $F$ must cancel every root of $P$ with its full multiplicity.

For a determinant-one lift of a hyperbolic holonomy representation, the squared matrix trace of each fixed group word is a polynomial in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of affine algebraic sets.

A polynomial equation has the form $p(x_1,\dots ,x_n)=0$ for a polynomial $p$. Its solutions are the points at which that polynomial vanishes.

Polynomial evaluation replaces the indeterminate of a polynomial by an input value. For a fixed scalar $a$, the rule $E_a:p\mapsto p(a)$ is a linear functional on a vector space of polynomials, because evaluating a linear combination gives the same linear combination of the values. For $p(t)=1+2t+3t^2$, evaluation at zero gives one and evaluation at two gives seventeen. Polynomial interpolation uses these measurements to reconstruct a polynomial subject to a polynomial degree bound.

A polynomial factor $g$ of a polynomial $f$ satisfies $f=gh$ for another polynomial $h$. Common polynomial factors are cancelled when expressing a rational function in reduced form; its degree of a rational map of the Riemann sphere is determined after this cancellation.

A polynomial function is the function obtained by evaluating a polynomial on a specified ring or field. Distinct polynomials can induce the same polynomial function over a finite field; for example, $x^p$ and $x$ agree on ${\mathbb{F}}_p$.

Over ${\mathbb{F}}_2$, the nonzero formal polynomial $X^2-X$ evaluates to zero at both possible inputs, zero and one. It therefore induces the same polynomial function as the zero polynomial, despite having different coefficients and polynomial degree. Over an infinite field, equality at every input forces equality of polynomials by the root bound for a polynomial. The distinction matters whenever a representation is supposed to identify the function uniquely.

For a compact set $K\subset {\mathbb{C}}^n$, its polynomial hull is the set displayed, where $p$ ranges over all complex multivariate polynomials. It contains $K$. In one complex dimension, bounded complementary components are filled in: the maximum modulus principle bounds every polynomial inside those components by its boundary values. Polynomial approximation and separation distinguish points outside the hull.

A polynomial identity is equality of two polynomials for every value of the variables, rather than an equation imposed only at selected roots. Over the real or complex numbers, this is equivalent to equality of corresponding coefficients: their difference, if nonzero, has only finitely many roots in one variable. The binomial theorem is an example. Coefficient comparison transfers a polynomial product identity into finite sums of binomial coefficients.

Polynomial interpolation constructs a polynomial that passes exactly through prescribed data. Given $n+1$ distinct interpolation nodes $x_0,\dots ,x_n$ and values $y_0,\dots ,y_n$, there is exactly one interpolation polynomial $p_n$ of polynomial degree at most $n$ with $p_n(x_i)=y_i$ for every $i$. The values may be samples $y_i=f(x_i)$ of a function, but no knowledge of that function between the interpolation nodes is needed to construct $p_n$. The Lagrange polynomial and Newton polynomial are two representations of this same answer.^[ref]

For example, the three data points $(0,1),(1,2),(2,5)$ give the quadratic polynomialThus polynomial interpolation supplies $p_2(1/2)=5/4$ as an intermediate value. Whether this approximates an unknown underlying function well is a separate question about polynomial interpolation error.

Both hypotheses in the uniqueness statement matter. The Lagrange cardinal polynomials prove existence constructively. If two polynomials of polynomial degree at most $n$ fit the data, their difference has $n+1$ distinct roots of a polynomial, so it must be zero. Repeated interpolation nodes with conflicting values have no solution, and repeated equal values provide no additional independent condition; prescribed derivatives instead lead to Hermite interpolation. Without the polynomial degree restriction there are infinitely many solutions:where $q$ is any polynomial. By the factor theorem, this describes every polynomial fitting the same data. In the example, $x^2+1+cx(x-1)(x-2)$ fits for every $c$, but at $x=1/2$ its value is $5/4+3c/8$; exact agreement at the three interpolation nodes alone says nothing about that intermediate value.

The usefulness of polynomial interpolation is that data become an explicit function whose values, derivatives, and integrals can be calculated. Differentiating or integrating the Lagrange cardinal polynomials produces differentiation weights from nodal evaluations or a quadrature rule. Exact passage through observations is the defining constraint. In polynomial regression, linear least squares instead chooses a polynomial degree and minimizes a sum of squared residuals, generally without passing through every observation. Interpolating noisy values reproduces their errors at the interpolation nodes and can amplify them between interpolation nodes; polynomial interpolation does not itself smooth measurement noise. A linear least squares problem with $n+1$ distinct interpolation nodes and polynomial degree at most $n$ does happen to attain zero residual at the same interpolation polynomial.

To divide $q$ by a nonzero polynomial $Q$ over a field, subtract the multiple of $Q$ that cancels the leading term of the current remainder, and add that multiple to the quotient. Each step strictly decreases the remainder degree, so the process terminates with $q=Qs+r$ and $\mathrm{deg}r<\mathrm{deg}Q$. If two such pairs existed, $Q(s-s\prime )=r\prime -r$ would have degree both at least $\mathrm{deg}Q$ and less than $\mathrm{deg}Q$, unless both differences vanished.

After $T$ input-bit queries, every probability amplitude is a polynomial of degree at most $T$, because the bit-query matrix entries are affine in the input bits. An acceptance probability is a sum of squared absolute amplitudes and has polynomial degree at most $2T$. Multilinear reduction on the Boolean cube does not increase degree. Exact acceptance therefore implies $Q_E(f)\ge \lceil \mathrm{deg}(f)/2\rceil$. For bounded error the acceptance polynomial only approximates the Boolean function; its exact representing degree is not the relevant bound.