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Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii Condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated with some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii Condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LP or ULP condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULP) and the books by R. Sakamoto (LP) or by S. Benzoni-Gavage and myself (ULP).

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii Condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.

For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.

A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).

A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called Lopatinskii condition, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied uniformly. In several interesting cases, LP or ULP condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called dissipative symmetrizer, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULP) and the books by R. Sakamoto (LP) or by S. Benzoni-Gavage and myself (ULP).