Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem is also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem.

In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\poly(\log q, d, p, n)\right) \, ,$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.

Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications.

\begin{itemize}
\item We show that there is an \emph{algebraic} data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Milterson \cite{M95} who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity.

\item We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program \cite{Valiant1977}. More precisely, for every fixed prime $p$, we show that for every constant $\epsilon > 0$, and large enough $n$, the rank of any $n \times n$ Vandermonde matrix $V$ over the field $\F_{p^a}$ can be reduced to $
\left(n/\exp(\Omega(\poly(\epsilon)\sqrt{\log n}))\right) $ by changing at most $n^{\Theta(\epsilon)}$ entries in every row of $V$, provided $a \leq \poly(\log n)$. Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression \cite{DL20}.

\end{itemize}

Fixed reference issues

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem is also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem.

In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\poly(\log q, d, p, n)\right) \, ,$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.

Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications.

\begin{itemize}
\item We show that there is an \emph{algebraic} data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Milterson \cite{M95} who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity.

\item We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program \cite{Valiant1977}. More precisely, for every fixed prime $p$, we show that for every constant $\epsilon > 0$, and large enough $n$, the rank of any $n \times n$ Vandermonde matrix $V$ over the field $\F_{p^a}$ can be reduced to $
\left(n/\exp(\Omega(\poly(\epsilon)\sqrt{\log n}))\right) $ by changing at most $n^{\Theta(\epsilon)}$ entries in every row of $V$, provided $a \leq \poly(\log n)$. Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression \cite{DL20}.

\end{itemize}

Fixed some typos.

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem is also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem.

In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\poly(\log q, d, p, n)\right) \, ,$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.

Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications.

\begin{itemize}
\item We show that there is an \emph{algebraic} data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Milterson \cite{M95} who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity.

\item We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program \cite{Valiant1977}. More precisely, for every fixed prime $p$, we show that for every constant $\epsilon > 0$, and large enough $n$, the rank of any $n \times n$ Vandermonde matrix $V$ over the field $\F_{p^a}$ can be reduced to $
\left(n/\exp(\Omega(\poly(\epsilon)\sqrt{\log n}))\right) $ by changing at most $n^{\Theta(\epsilon)}$ entries in every row of $V$, provided $a \leq \poly(\log n)$. Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression \cite{DL20}.

\end{itemize}

Fixed some typos.

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem is also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem.

In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\poly(\log q, d, p, n)\right) \, ,$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.

Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications.

\begin{itemize}
\item We show that there is an \emph{algebraic} data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Milterson \cite{M95} who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity.

\item We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program \cite{Valiant1977}. More precisely, for every fixed prime $p$, we show that for every constant $\epsilon > 0$, and large enough $n$, the rank of any $n \times n$ Vandermonde matrix $V$ over the field $\F_{p^a}$ can be reduced to $
\left(n/\exp(\Omega(\poly(\epsilon)\sqrt{\log n}))\right) $ by changing at most $n^{\Theta(\epsilon)}$ entries in every row of $V$, provided $a \leq \poly(\log n)$. Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression \cite{DL20}.

\end{itemize}