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Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$, where the second derivative is understood in the sense of generalized functions.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities of algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$, where the second derivative is understood in the sense of generalized functions.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities of algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

Bernstein has constructed a module over the ring of differential operators (the module has a formal generator $|f|^\lambda$) for which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities in algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$, where the second derivative is understood in the sense of generalized functions.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities of algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Here is an application to (functional) analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities in algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.

Here is an application of the Hironaka's resolution of singularities theorem to functional analysis. In 1963 I. M. Gelfand posed the following problem. Given a polynomial $f$ on $\mathbb{R}^n$. For a complex parameter $\lambda$ the power $|f|^\lambda$ is a continuous function if $Re(\lambda)> 0$. Gelfand's question was whether $|f|^\lambda$ can be meromorphically continued in the parameter $\lambda$ to the whole complex plane as a generalized function on $\mathbb{R}^n$.

(Example: on $\mathbb{R}$ the meromorphic continuation of the function $|x|^\lambda$ to $\lambda=-1$ has a pole, and to $\lambda=-2$ equals to $(ln|x|)''$.)

To the best of my knowledge, the first complete positive solution of this problem was obtained by J. Bernstein and S. Gelfand (1969) and independently by M. Atiyah (1970). They used the Hironaka resolution of singularities in algebraic varieties. The latter result is purely algebro-geometric and very difficult (Hironaka was awarded the Fields medal in 1970 for this result).

Let me also mention that in 1972 J. Bernstein invented another approach to prove the above result without using the Hironaka theorem. This approach is also purely algebraic, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-a-cont-FAN.pdf It has far reaching extensions. The main step was to show that there exists a differential operator $D_\lambda$ whose coefficients depend polynomially on the coordinates in $\mathbb{R}^n$ and rationally on $\lambda$ such that $D_\lambda(|f|^{\lambda+1})=|f|^\lambda$. Using this formula recursively, one extends the distribution from the half plane $Re(\lambda)>0$ to the whole complex plane.

The method developed by Bernstein was purely algebraic: out of $|f|^\lambda$ he constructed a module over the ring of differential operators about which he had to prove several things, mainly that it is holonomic. This method became most important in Bernstein's subsequent approach to the theory of algebraic D-modules.