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Hilbert's $16^{\rm th}$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure great mathematical figure.

Hilbert's $16^{\rm th}$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure.

Hilbert's $16^{\rm th}$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure.

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answered Dec 16, 2012 at 15:26

Rodrigo A. Pérez

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Hilbert's $16^{\rm th}$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure.