<PAGE>

                                                                    EXHIBIT 31.1

                          RULE 13a-14(a) CERTIFICATION

I, Frank C. Lin, Chief Executive Officer of Trident Microsystems, Inc. (the
registrant), certify that:

1. I have reviewed this Annual Report on Form 10-K of Trident Microsystems, Inc.
for the fiscal year ended June 30, 2004 (including this Amendment to Annual
Report on Form 10-K/A);

2. Based on my knowledge, this report does not contain any untrue statement of a
material fact or omit to state a material fact necessary to make the statements
made, in light of the circumstances under which such statements were made, not
misleading with respect to the period covered by this report;

3. Based on my knowledge, the financial statements, and other financial
information included in this report, fairly present in all material respects the
financial condition, results of operations and cash flows of the registrant as
of, and for, the periods presented in this report;

4. The registrant's other certifying officer and I are responsible for
establishing and maintaining disclosure controls and procedures (as defined in
Exchange Act Rules 13a-15(e) and 15d-15(e)) for the registrant and have:

      (a) Designed such disclosure controls and procedures, or caused such
      disclosure controls and procedures to be designed under our supervision,
      to ensure that material information relating to the registrant, including
      its consolidated subsidiaries, is made known to us by others within those
      entities, particularly during the period in which this report is being
      prepared;

      (b) Evaluated the effectiveness of the registrant's disclosure controls
      and procedures and presented in this report our conclusions about the
      effectiveness of the disclosure controls and procedures, as of the end of
      the period covered by this report based on such evaluation; and

      (c) Disclosed in this report any change in the registrant's internal
      control over financial reporting that occurred during the registrant's
      most recent fiscal quarter (the registrant's fourth fiscal quarter in the
      case of an annual report) that has materially affected, or is reasonably
      likely to materially affect, the registrant's internal control over
      financial reporting; and

5. The registrant's other certifying officer and I have disclosed, based on our
most recent evaluation of internal control over financial reporting, to the
registrant's auditors and the audit committee of the registrant's board of
directors (or persons performing the equivalent functions):

      (a) All significant deficiencies and material weaknesses in the design or
      operation of internal control over financial reporting which are
      reasonably likely to adversely affect the registrant's ability to record,
      process, summarize and report financial information; and

      (b) Any fraud, whether or not material, that involves management or other
      employees who have a significant role in the registrant's internal control
      over financial reporting.

      Dated:  October 28, 2004                      /s/  FRANK C. LIN
                                                    -----------------
                                                         Frank C. Lin
                                                         Chief Executive Officer