Polynomial interpolation over fields

Choose n distinct nodes in a field and label their desired values. For a particular node, multiply x minus every other node, producing a polynomial that vanishes at all the other locations. Divide by the value of that product at the selected node so that the selected value becomes one. Every denominator factor is the difference of two distinct field elements, hence is nonzero and invertible. This constructs a Lagrange basis polynomial of degree n minus one.

Multiply each basis polynomial by its prescribed value and add them. At a selected node, all but one summand vanish and the remaining summand gives exactly the desired value. This proves existence without solving a coefficient system. The construction works over any field with enough distinct elements for the nodes, including a finite field. Over a general ring, distinct nodes alone are insufficient because their differences may be nonunits.

If two polynomials of degree below n meet the same n prescriptions, their difference has degree below n and vanishes at all n nodes. A nonzero polynomial over a field cannot have more roots than its degree, so the difference is zero. The degree restriction is essential. Adding any multiple of the product of x minus all nodes gives another polynomial with exactly the same prescribed values, generally at higher degree.

This is also a quotient-ring statement. The simultaneous evaluation map has kernel generated by that node product, since the distinct linear ideals are pairwise comaximal. Its target is the product of n copies of the field, and the CRT makes it surjective. Polynomial division then selects a unique representative of degree below n for each quotient class. Lagrange interpolation provides an explicit inverse from the value tuple to that representative.

Over the reals, the theorem guarantees exact interpolation for exact inputs, but it does not guarantee numerical stability. Closely spaced nodes can make basis denominators small, amplifying roundoff or measurement noise in some representations. A different basis or node choice can change computational behavior without changing the unique exact polynomial. These numerical questions require error analysis beyond the existence proof, and should not be dismissed simply because a symbolic formula is available.

Likewise, matching observations does not prove that the interpolating polynomial describes the mechanism between nodes or outside the sampled interval. In a finite field, different formal polynomials can even define the same function at every field element: the node product over the whole field vanishes everywhere. Keep three statements separate: a unique low-degree interpolant exists under the hypotheses, an algorithm computes an approximation to it, and a chosen model predicts useful new observations. Only the first follows from this algebraic theorem.