Polynomial Interpolation of Power Functions

• Polynomial interpolation of power functions is a method to recover or approximate expressions like f(x)=x^k from discrete data using specialized algorithms.
• Key approaches include black-box modular evaluations, Taylor expansion, sparse shifted-lacunary recovery, and spline patching to manage complexity and ensure accuracy.
• These techniques deliver output-sensitive performance and have practical applications in computational algebra, numerical analysis, and theoretical computer science.

Polynomial interpolation of power functions encompasses the recovery or approximate representation of expressions of the form $f(x) = x^k$, or more generally, $f(x) = \sum_i c_i(x-\alpha)^{e_i}$, from discrete data or black-box evaluations. Specialized algorithms and algebraic constructions exploit the sparsity, structure, or specific evaluation context of power functions to achieve output-sensitive complexity, exact recovery, or controlled approximation error. Methods range from Taylor-based local expansion, modular interpolation over finite fields, sparse shifted-lacunary reconstruction, and spline-based patching for large domains. The following sections survey the principal approaches, key theorems, complexity bounds, and algorithmic paradigms for interpolation of power functions.

1. Frameworks for Power Function Interpolation

Polynomial interpolation for power functions is set in several distinct frameworks, each tailored to model constraints and objectives:

• Black-Box Rational Interpolation: Evaluation of an unknown $f(x)\in\mathbb{Q}[x]$ is performed modulo prime $p$ via a black box; the goal is to recover the sparsest shifted-power basis representation $f(x) = \sum_{i=0}^{t} c_i (x-\alpha)^{e_i}$ with rational coefficients $c_i$ and shift $\alpha$ (0810.5685).
• Direct Taylor Expansion: Given $N=2\alpha+1$ equispaced evaluations $y_k$ of a smooth function (specifically power functions), construct the interpolant as a Taylor polynomial about the midpoint $x_0$ by solving a confluent Vandermonde system for the derivatives $D_m$ (Shukurov, 2020).
• Finite Field Oracle Access: For $f(x)\in\mathbb{F}_q[x]$, only access to $f(x)^e$ (for fixed $e\mid q-1$) at points in $\mathbb{F}_q$ is permitted; the problem is to interpolate $f$ or a polynomial $g$ such that $g^e \equiv f^e$ modulo $X^q-X$ (Ivanyos et al., 2015).
• Spline Patch Approximation: Piecewise polynomial splines are constructed to approximate $X^j$ by rearranging Faulhaber’s formula for sums of powers, explicitly providing a polynomial $P(m,X,N)$ that matches derivatives up to order $m$ at $X=N$, and forming splines to cover large intervals with bounded relative error (Kolosov, 24 Feb 2025).

2. Algorithmic Strategies and Explicit Constructions

Shifted-Lacunary Interpolation

Sparse recovery proceeds in two main phases (0810.5685):

1. Sparsest Shift Computation: Identify the minimal shift $\alpha$ so that $f(x+\alpha)$ is $t$-sparse. For sufficiently large degree, $\alpha$ is unique. This is effected by evaluating modular images $f^{(p)}(x)$ for several primes $p$ and using CRT (Chinese Remainder Theorem) for rational reconstruction.
2. Sparse Power Recovery: Given the shift, perform interpolation for $g(x)=f(x+\alpha)$ in the power basis, extract exponents $e_i$ via symmetric polynomial construction, and reconstruct coefficients $c_i$ via modular projections and CRT.

Taylor Polynomial Interpolation

For power functions $f(x)=x^n$, the interpolation polynomial $P(x)$ of degree $N-1$ centered at $x_0$ uses central difference formulas for derivatives $D_m$, assembled from the data and then combined into $P(x)=\sum_{m=0}^{N-1}(D_m/m!)(x-x_0)^m$. For $n<N-1$, $P(x)$ matches $f(x)$ exactly everywhere (Shukurov, 2020).

Finite Field High Power Oracle Interpolation

Three algorithmic families address interpolation from high powers (Ivanyos et al., 2015):

• Naive Black-Box: Interpolate $G(x)=f(x)^e$ as a dense polynomial of degree $ed$ given $ed+1$ queries, then factor $G$.
• Improved Deterministic: Uses algebraic-combinatorial methods, Nullstellensatz reductions, product-set growth, and character sum estimates to reduce query complexity asymptotically to $e^{o(1)}$ in certain regimes.
• Randomized/Quantum: Exploits Weil-bound, small sample distinguishing sets, Grover's search (quantum), and discrete log computations to reduce query number to $O(d\log q)$, though the time complexity is exponential in $d$.

Spline Approximation for Power Functions

Spline construction for $X^j$ utilizes the polynomial $P(m,X,N)$, derived through explicit expansion from Faulhaber’s identity and matched at $X=N$ up to $m$ derivatives. These splines are organized over overlapping intervals of width $2R$, where $R$ is chosen such that the relative error $\mathrm{RelErr}(X)\leq\varepsilon$. For arbitrary exponents, the piecewise splines take the form $S_k(X)$ with appropriate degree and matching at segment knots for error control (Kolosov, 24 Feb 2025).

3. Theoretical Bounds and Complexity

Uniqueness and Minimality

For shifted-lacunary interpolation, if $\deg f \ge 2t+1$, the shift $\alpha$ yielding $t$-term sparsity is unique; $t$ is absolutely minimal over all shifted bases (0810.5685).

Error Estimates

Taylor-based and spline-based interpolation for $f(x)=x^n$ yields exact reconstruction for $n\leq N-1$. Beyond that, the error for $N$-point interpolation grows as

$$|R_{N-1}(x)| \leq \frac{n!}{(n-N)!}\cdot\frac{\max_{\xi\in[x_0,x]}|\xi|^{n-N}}{N!}|x-x_0|^N$$

For splines, the relative error over a segment centered at $N$ is

$$\mathrm{RelErr}(X) \leq \binom{2m+1}{m} \left( \frac{|X-N|}{N} \right)^{m+1}$$

with interval half-width $$R = \left( \frac{\varepsilon}{\binom{2m+1}{m}} \right)^{1/(m+1)} N$$ for error threshold $\varepsilon$ (Kolosov, 24 Feb 2025).

Bit-Complexity

Shifted-lacunary algorithms exhibit polynomial bit-complexity in $t$, $\log\deg f$, $\max_i\mathrm{size}(c_i)$, and $\mathrm{size}(\alpha)$, with output-sensitive performance for high-degree, sparse polynomials (0810.5685).

Finite field high-power algorithms achieve deterministic query complexity $O(e^{o(1)})$ or $O(d\log q)$ in the randomized/quantum regime, with time exponential in $d$ (Ivanyos et al., 2015).

4. Explicit Formulas and Pseudocode Paradigms

Shifted-Lacunary Black-Box Interpolation (0810.5685)

Algorithm InterpolateShifted(f-blackbox, B_T,B_N,B_H,B_A): while size(P)<2^{B_A+1}: p ← nextPrime() interpolate f^{(p)} if deg f^{(p)}≥2B_T+1: for γ=0…p−1: sparsity s(γ)=#terms in f^{(p)}(x+γ) α_p←arg min_γ s(γ) record α mod p=α_p α←RationalReconstruct({(p,α_p)}) For g(x)=f(x+α): sparse interpolation

Taylor Polynomial Construction for Power Functions (Shukurov, 2020)

Given $N=2\alpha+1$ data points $y_k=f(x_0+k\Delta x)$,

• Construct confluent Vandermonde matrix $C_{kj}=(k\Delta x)^{j}/j!$
• Solve $C\cdot D=Y$ to obtain $D_m=f^{(m)}(x_0)$ (via central differences for equispaced points)
• Set $P(x)=\sum_{m=0}^{N-1}\frac{D_m}{m!}(x-x_0)^m$

Spline Construction for Power Functions (Kolosov, 24 Feb 2025)

• For target exponent $j$, decompose $j=2m+1$ or $j=2m,2m+2$
• On interval $[A,B]$, set knots at $N_0=A,N_1,A+L,\dots,N_K=B$
• For each segment, polynomial $S_k(X)=P(m,X,N_k)$ (or appropriate variant)
• Enforce $C^0$ continuity at segment boundaries
• Ensure error $\leq\varepsilon$ on each segment by proper choice of radii $R_k$

5. Comparative Analysis of Methodologies

Approach	Complexity	Error Control	Applicability
Shifted-Lacunary Black-Box	Poly in size(f)	Exact	Sparse, high-degree polynomials
Taylor Series Interpolation	Poly in $N$	Exact for degree$<N$	Smooth or polynomial power functions
Finite Field High-Power Polynomial	$e^{o(1)}$, $O(d\log q)$	Probabilistic	$\mathbb{F}_q$, powers, quantum
Spline Approximation	$O(Km)$, $K\propto (B-A)/R$	Controlled, $<1\%$	Large intervals, arbitrary exponents

Shifted-lacunary and finite field methods excel when algebraic sparsity, black-box modular access, or field constraints apply. Taylor and spline interpolants provide powerful and exact recovery for polynomial powers within degree or segment limits, and spline constructions are effective for large domains with explicit error bounds.

6. Extensions, Open Problems, and Applications

Research has suggested several avenues for development:

• Multivariate Shifted-Sparsity: Generalize methods to polynomials with unknown shifts in multiple variables.
• Disjoint Shifts: Interpolate polynomials represented as sums of shifted power functions with distinct shifts.
• Circuit-Size Recovery: Formulate interpolation targeting minimal formula or circuit size representations (open problem) (0810.5685).
• Spline Adaptivity: Use adaptive splines or moving midpoints to control error amplification for large-increment extrapolation (Kolosov, 24 Feb 2025).
• Quantum Algorithms in Finite Fields: Further optimize sampling strategies and quantum search over root ambiguities for high-degree polynomials (Ivanyos et al., 2015).
• Domain Decomposition: Piecewise Taylor or spline methods to extend accurate polynomial interpolation over large intervals (Shukurov, 2020).

A plausible implication is that advances in sparse shifted-power and modular interpolation increasingly enable recovery and efficient approximation of power functions even in settings with astronomically high degree or stringent data access constraints. The algorithmic principles connect to both classical symbolic computation and emerging quantum search paradigms, informing computational algebra, numerical analysis, and theoretical computer science.