Expose fixed-size Array inner product in Polars SQL

[0guban0v]

Description

Follow-up to #28504

Polars provides native row-wise dot product for fixed-size Array expressions:

import polars as pl

df = pl.DataFrame(
    {
        "lhs": [[1.0, 2.0], [3.0, 4.0]],
        "rhs": [[10.0, 20.0], [30.0, 40.0]],
    },
    schema={
        "lhs": pl.Array(pl.Float64, 2),
        "rhs": pl.Array(pl.Float64, 2),
    },
)

print(df.select(dot=pl.col("lhs").arr.dot("rhs")))

Output:

shape: (2, 1)
┌───────┐
│ dot   │
│ ---   │
│ f64   │
╞═══════╡
│ 50.0  │
│ 250.0 │
└───────┘

Equivalent operation is not exposed in Polars SQL:

df.sql("SELECT ARRAY_INNER_PRODUCT(lhs, rhs) AS dot FROM self")

Expected

Expose native fixed-size Array dot product as ARRAY_INNER_PRODUCT(lhs, rhs) with ARRAY_DOT_PRODUCT as alias. Both names follow existing SQL precedent in DuckDB here

For two fixed-size Array expressions, SQL function should lower directly to lhs.arr.dot(rhs) and inherit existing supertype coercion, broadcasting, null handling, output dtype, integer overflow, and execution-engine behavior.
It should not expand into (lhs * rhs).arr.sum(), which materializes an intermediate product Array.

Known-width SQL literals should also be supported:

SELECT ARRAY_INNER_PRODUCT(lhs, [10.0, 20.0]) AS dot FROM self

Polars SQL currently lowers [10.0, 20.0] to List(Float64). We need to infer inner dtype for conversion to fixed-size Array that would result one-row Array then use existing broadcasting.