Polynomial function
A polynomial function is a function expressible as a finite sum of constant multiples of nonnegative integer powers of one or more variables.
Geometry for Programmers
A polynomial is an expression formed from sums of variable powers multiplied by coefficients.
- How it is used
- Supports approximation, interpolation, and curve construction for geometric modeling.
- How deep it goes
- Defines forms, degree, roots, derivatives, efficient evaluation, and coefficient-based curve constraints without deriving numerical error analysis.
- Examples
- Uses cubic, quadratic, and linear graphs, plus a cubic fitted to two points and two tangent directions.
- What you can do after
- Apply it via coefficient forms and Horner’s scheme to evaluate polynomials and formulate interpolation or tangent constraints.
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Elliptic Curve Cryptography for Developers
A polynomial is an algebraic expression formed from coefficients, variables, and nonnegative integer powers combined by addition and multiplication.
- How it is used
- Supports finite-field extension arithmetic used by elliptic-curve pairing routines.
- How deep it goes
- Defines representation and utilities, then details modular addition and subtraction while deferring field-extension details.
- Examples
- Uses a p=1187 degree-three extension, a modular polynomial-sum exercise, and C polynomial routines.
- What you can do after
- Build it from scratch as fixed-capacity C structures with initialization, cleanup, arithmetic, comparison, printing, and random generation.
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Real-World Cryptography
A polynomial is a mathematical expression formed from coefficients and nonnegative integer powers of a variable.
- How it is used
- Encodes several execution constraints as one polynomial for zk-SNARK proving.
- How deep it goes
- Builds polynomials from roots, selector functions, coefficients, and hidden-point evaluations.
- Examples
- Uses roots 1, 2, and 3 with equations involving a, b, m, v, and w.
- What you can do after
- Evaluate/understand how polynomial roots represent correct program constraints while hiding selected values.
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Related concepts
- cubic polynomialincludes · A cubic polynomial is a polynomial of degree three.
- polynomial degreemeasured by · The degree of a polynomial is the greatest exponent with a nonzero coefficient.
- mathematical functionkind of · A mathematical function is a mapping that assigns exactly one output to each permitted input in its domain.
- Bézier curveincludes · A Bézier curve is a parametric curve whose shape is determined by control points through Bernstein basis functions.
- derivativeproduces · A derivative is a mathematical quantity that measures the instantaneous rate of change of a function with respect to an input variable.
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