ring_poly_mul

std.algebra.ring_poly_mul · Level L3

The product of two polynomials over a finite commutative ring, by finite convolution with the ring's own tables: (pq)_k = Σ_{i+j=k} p_i·q_j, the sum taken with the addition table. A Scan of ring_poly_step over the coefficients of p. With zero divisors the degree of a product can fall below the sum of the degrees (over Z/4, (2x + 1)(2x + 1) = 1), so the degree formula is not used. R[x] is not a finite ring: when 1 ≠ 0 the powers 1, x, x², … are all distinct.

(pq)_k = Σ_{i+j=k} p_i q_j

Signature

ring_poly_mul(Ad: i64[n, n], Mu: i64[n, n], p: i64[a], q: i64[c]) → i64[a+c−1]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the double loop over p_i and q_j, with the ring's tables in exact rational arithmetic, on all 40 test cases.
All 285 int64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Identity

Calls

group_axioms ring_poly_step

Called by

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sha256:e08c14f1c7a580a5d873160cd9d7cc091dd12d416e7122f1050342ab8354d578

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.ring_poly_mul · Ω:ring_poly_mul
Pins: sha256:dae4c01255ab227c49504d7409b6233c48324b20283cc7d8e1a2c6bf724d7016this graph and the 2 it reaches through calls
Evidence: sha256:ea569ee7e3ab69f57e036bcb4fa350f91b968938a99b5c6059b731014338fc3fthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.