Work with callable polynomials

Load the focused polynomial plugin directly, or load .ratfun/.algebra, which require and activate it automatically.

Runnable RiXRun cell

.Plugin.Load("poly"); P := `x^3 - 6x^2 + 11x - 6`.P(); {: P(2), P.Coefficients(), P.Degree() };

Contextual coefficients

A symbolic specification declares the indeterminate explicitly. Other ordinary names remain live closure coefficients.

Runnable RiXRun cell

.Plugin.Load("poly"); y := 2; P := {#x# x^2 + y*x}.P(); y ~= 3; P(2); ## 10

Reactive dependency chains

Structural-arithmetic splices take snapshots. Placing a reactive read inside a reactive polynomial definition records the dependency and rebuilds the value. Derived reactive polynomials record their $P reads in the same graph.

Runnable RiXRun cell

.Plugin.Load("poly"); $$y := 2; $$P := `.p.Var(x):x^2 + @($y)*x`; $$Q := $P*$P + 1; $y := 3; {: $P(2), $Q(2) }; ## (10, 101)

Operators and composition

Polynomial addition, subtraction, multiplication, nonnegative integral powers, scalar division, negation, and equality preserve Polynomial identity. Passing a Polynomial to another Polynomial evaluates by substitution, so it is ordinary composition.

Runnable RiXRun cell

.Plugin.Load("poly"); P := `x^2 + 1`.P(); Q := `x + 2`.P(); R := P(Q); A := P + 2; B := Q^3; {: R(3), A(3), B(1) };

The Polynomial itself owns exact receiver methods and quotient/remainder operators. Loading .algebra also loads .ratfun and adds the presentation wrapper used for verified division metadata and synthetic-division Grids:

Runnable RiXRun cell

.Plugin.Load("algebra"); ## also loads poly P := .p`x^3 - 6x^2 + 11x - 6`; F := .p`x - 2`; division := P.SyntheticDiv(2); Q := P // F; ## quotient R := P % F; ## remainder pair := P /% F; ## {: quotient, remainder } fraction := P/F; ## canonical RationalFunction from .ratfun {: division.Quotient()(4), Q(4), R(4), pair[1](4) };

Only reactive definitions recompute. Q := $P*$P + 1 takes the current value of $P; $$Q := $P*$P + 1 records a dependency and rebuilds when $P changes. The same rule applies to every other RiX value.

Verify exact factors and multiplicities

Phase 2 algorithms return canonical polynomials or versioned evidence records. The checks are performed with exact rational arithmetic.

Runnable RiXRun cell

.Plugin.Load("poly"); P := .p`(x-1)^2*(x+2)^3`; support := .p`(x-1)*(x+2)`; evidence := P.FactorEvidence(); {= gcd=P.Gcd(support).Coefficients(), squareFree=P.SquareFreeDecomposition(), roots=evidence.Roots(), verified=evidence[:verified], resultant=P.Resultant(.p`x+3`), sign=P.SignEvidence(0), rootCount=P.RootCountEvidence(-10:10) };