Definition. Univariate polynomial maps into a vector space [cn-0064]
Let \(W\) be a real vector space of dimension \(N\). Here \(W\) is the ambient space of polynomial values. Choose a basis \(e_1,\ldots ,e_N\). A univariate polynomial map \(P:\mathbb R\to W\) has one scalar polynomial for each coordinate of its value:
\[P(t)=\sum _{j=1}^{N}p_j(t)e_j,\qquad p_j\in \mathbb R[\xi ].\]This definition is independent of the chosen basis: a change of coordinates takes real linear combinations of the scalar polynomials \(p_j\). For example, \(P(t)=(1+2t,3-t^2)\) is a polynomial map from \(\mathbb R\) to \(\mathbb R^2\). Its two coordinates are two scalar polynomial functions of the same variable.
Writing \(p_j(\xi )=\sum _{r=0}^{h}c_{jr}\xi ^r\) and collecting equal powers expresses the map in terms of its vector coefficients \(w_r\):
\[P(\xi )=\sum _{r=0}^{h}w_r\xi ^r,\qquad w_r=\sum _{j=1}^{N}c_{jr}e_j\in W.\]The paper's notation \(W[\xi ]\) encodes these maps by their coefficient expansions. For a subspace \(Z\subseteq W\), \(Z[\xi ]\) consists of the maps with every \(w_r\in Z\), so their values also lie in \(Z\). For \(P\ne 0\), its degree is the largest \(r\) with \(w_r\ne 0\); the zero map is included in every degree bound.
For degree at most \(h\), the coefficient tuple \((w_0,\ldots ,w_h)\) belongs to \(W^{h+1}\). Slot \(r\) corresponds to monomial \(\xi ^r\). Thus \(W[\xi ]_{\leq h}\cong W^{h+1}\). The polynomial coefficient matrix \(C=(c_{jr})\) has \(N\) rows, one for each coordinate function \(p_j\), and \(h+1\) columns, one for each monomial \(1,\xi ,\ldots ,\xi ^h\).
\[[P(\zeta )]_{(e_1,\ldots ,e_N)}=C\begin {pmatrix}1\\\zeta \\\vdots \\\zeta ^h\end {pmatrix}.\]The distinction is between the coordinate index \(j\) of a value in \(W\) and the degree index \(r\) within each scalar polynomial \(p_j\). The monomial correspondence is with \(r\); the dimension \(N\) of the value space and the polynomial degree bound \(h\) are independent.
For example, with \(W=\mathbb R^2\),
\[P(\xi )=(1,3)+\xi (2,0)+\xi ^2(0,-1)=(1+2\xi ,\,3-\xi ^2).\]Its coefficient matrix has columns corresponding, in order, to \(1,\xi ,\xi ^2\):
\[C=\begin {pmatrix}1&2&0\\3&0&-1\end {pmatrix}.\]This is a degree-two polynomial map \(\mathbb R\to \mathbb R^2\); for instance \(P(1)=(3,2)\).
In the rank-sum application [cn-0027], \(W=\mathbb R^d\oplus \mathbb R^d\). A polynomial map has the form \(P(t)=(a(t),b(t))\), where \(a,b\) are \(\mathbb R^d\)-valued polynomial maps. Requiring \(P(t_i)\) to lie in the graph [cn-0026] of \(X_i\) means \(b(t_i)=X_i a(t_i)\). This explains why evaluation in the subspace argument must produce a vector in \(W\).
These auxiliary maps serve the interpolation proof; they do not approximate drawing arcs. The same definition works over any infinite field by replacing \(\mathbb R\) throughout. Infinitude ensures that a scalar polynomial is determined by its values, so degree and coefficients are well-defined for the map. See the field-of-scalars remark [cn-0065] for the algebraic inequality over arbitrary fields.
- Proof dependencies
- None.
- Provenance
- Notation used in OpenAI, 23 September 2026 [cn-0042], proof of Lemma 2.1; explanatory definition added on 9 October 2026.