From above, the term $\prod_{i=1}^n X_i^{d_i}$ is a {\em monomial}. A polynomial $\poly\inparen{\vct{X}}$ is in standard monomial basis (\abbrSMB) when we keep only the terms with $c_{\vct{d}}\ne0$ from \Cref{eq:sop-form}.
The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such that $c_{(d_1,\dots,d_n)}\ne0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
As an example, the degree of the polynomial $X^2+2XY^2+Y^2$ is $3$.
Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins to produce an output tuple.
%in any clause of the $\raPlus$ query that created it.
We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \bi (\ti) $\pdb$, and tuple $\tup$ such that $\poly\inparen{\vct{X}}=\apolyqdt\inparen{\vct{X}}.$
%\begin{Definition}[Modding with a set]\label{def:mod-set}
%Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
%\end{Definition}
%For example for a set of polynomials $S=\inset{X^2-X, Y^2-Y}$, taking the polynomial $2X^2 + 3XY - 2Y^2\mod S$ yields $2X+3XY-2Y$.
The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is the same as \Cref{def:reduced-poly} with the added constraint that all monomials with variables $X_{\block, i}, X_{\block, j}, i\neq j$ from the same block $\block$ are omitted.
Consider a $\abbrBIDB$ polynomial $\poly\inparen{\vct{X}}= X_{1, 1}X_{1, 2}+ X_{1, 2}X_{2, 1}^2$. Then by \Cref{def:reduced-bi-poly}, we have that $\rpoly\inparen{\vct{X}}= X_{1, 2}X_{2, 1}$. Next, we show why the reduced form is useful for our purposes.
%, (recall the constraint on tuples from the same block being disjoint in a \bi).% any monomial containing more than one tuple from a block has $0$ probability and can be ignored).
% Intuitively, $\rpoly(\textbf{X})$ is the \abbrSMB form of $\poly(\textbf{X})$ such that if any $X_j$ term has an exponent $e > 1$, it is reduced to $1$, i.e. $X_j^e\mapsto X_j$ for any $e > 1$.
%For probability distribution $\pd$, % and its corresponding probability mass function $\probOf$,
%the set of valid worlds $\valworlds$ consists of all the worlds with probability value greater than $0$; i.e., for random world variable vector $\vct{W}$
%We state additional equivalences between $\poly(\vct{X})$ and $\rpoly(\vct{X})$ in \Cref{app:subsec-pre-poly-rpoly} and \Cref{app:subsec-prop-q-qtilde}.
Let $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability distribution $\pdassign$ over $\vct{\randWorld}^\numvar$ (the all worlds set) is induced by the probability vector $\probAllTup=(\prob_1, \ldots, \prob_\numvar)$. As in \Cref{lem:tidb-reduce-poly} for \abbrTIDB, any \abbrBIDB-lineage polynomial $\poly(\vct{X})$ based on $\pdb$ and query $\query$ we have:
By \Cref{lem:exp-poly-rpoly} and linearity of expectation, the following corollary results.
%Note that in the preceding lemma, we have assigned $\vct{p}$
%%(introduced in \Cref{subsec:def-data})
%to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that when we replace each variable $X_i$ with its probability $\prob_i$ in the reduced form of a \bi-lineage polynomial and evaluate the resulting expression in $\mathbb{R}$, then the result is the expectation of the polynomial.
If $\poly$ is a \bi-lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $\expct\pbox{\poly}=\rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $\bigO{\size\inparen{\poly}}$, where $\size\inparen{\poly}$ (\Cref{def:size-depth}) is proportional to the total number of multiplication/addition operators in $\poly$.